10 88

10_87

10_89

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 88's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 88 at Knotilus!

Square quasi-symmetrical depiction.

Knot presentations

Planar diagram presentation	X4251 X20,14,1,13 X8394 X2,9,3,10 X14,7,15,8 X18,15,19,16 X12,6,13,5 X10,18,11,17 X16,12,17,11 X6,19,7,20
Gauss code	1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, -2
Dowker-Thistlethwaite code	4 8 12 14 2 16 20 18 10 6
Conway Notation	[.21.21]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 5, Braid index is 5		[{3, 10}, {11, 5}, {9, 4}, {10, 6}, {5, 8}, {6, 2}, {1, 3}, {7, 9}, {8, 12}, {2, 11}, {12, 7}, {4, 1}]

[edit Notes on presentations of 10 88]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 88"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X4251 X20,14,1,13 X8394 X2,9,3,10 X14,7,15,8 X18,15,19,16 X12,6,13,5 X10,18,11,17 X16,12,17,11 X6,19,7,20

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, -2

In[6]:=

DTCode[K]

Out[6]=

4 8 12 14 2 16 20 18 10 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[.21.21]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(5,\{-1,2,-1,-3,2,-3,2,4,-3,4\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 5, 10, 5 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 10}, {11, 5}, {9, 4}, {10, 6}, {5, 8}, {6, 2}, {1, 3}, {7, 9}, {8, 12}, {2, 11}, {12, 7}, {4, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 15.6466 A-Polynomial See Data:10 88/A-polynomial

[edit Notes for 10 88's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 10 88's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+8t^{2}-24t+35-24t^{-1}+8t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 101, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+13q^{2}-16q+17-16q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}-a^{4}z^{2}+2a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}-3z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+6z^{8}a^{-2}+12z^{8}+7a^{3}z^{7}+14az^{7}+14z^{7}a^{-1}+7z^{7}a^{-3}+4a^{4}z^{6}-2a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}a^{-4}-12z^{6}+a^{5}z^{5}-11a^{3}z^{5}-32az^{5}-32z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-10a^{2}z^{4}-10z^{4}a^{-2}-6z^{4}a^{-4}-8z^{4}-a^{5}z^{3}+6a^{3}z^{3}+19az^{3}+19z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+3a^{4}z^{2}+7a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}+8z^{2}-a^{3}z-4az-4za^{-1}-za^{-3}-a^{2}-a^{-2}-1$
The A2 invariant	$-q^{16}+q^{14}+2q^{12}-3q^{10}+3q^{8}-2q^{4}+3q^{2}-3+3q^{-2}-2q^{-4}+3q^{-8}-3q^{-10}+2q^{-12}+q^{-14}-q^{-16}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+15q^{72}-15q^{70}+4q^{68}+21q^{66}-53q^{64}+91q^{62}-111q^{60}+94q^{58}-33q^{56}-75q^{54}+204q^{52}-299q^{50}+314q^{48}-218q^{46}+18q^{44}+223q^{42}-417q^{40}+487q^{38}-379q^{36}+134q^{34}+154q^{32}-373q^{30}+418q^{28}-277q^{26}+21q^{24}+235q^{22}-365q^{20}+303q^{18}-66q^{16}-243q^{14}+490q^{12}-562q^{10}+415q^{8}-96q^{6}-286q^{4}+588q^{2}-699+588q^{-2}-286q^{-4}-96q^{-6}+415q^{-8}-562q^{-10}+490q^{-12}-243q^{-14}-66q^{-16}+303q^{-18}-365q^{-20}+235q^{-22}+21q^{-24}-277q^{-26}+418q^{-28}-373q^{-30}+154q^{-32}+134q^{-34}-379q^{-36}+487q^{-38}-417q^{-40}+223q^{-42}+18q^{-44}-218q^{-46}+314q^{-48}-299q^{-50}+204q^{-52}-75q^{-54}-33q^{-56}+94q^{-58}-111q^{-60}+91q^{-62}-53q^{-64}+21q^{-66}+4q^{-68}-15q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_88.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3q^{9}-4q^{7}+5q^{5}-3q^{3}+q+q^{-1}-3q^{-3}+5q^{-5}-4q^{-7}+3q^{-9}-q^{-11}$
2	$q^{32}-3q^{30}+11q^{26}-14q^{24}-10q^{22}+37q^{20}-18q^{18}-37q^{16}+54q^{14}-53q^{10}+36q^{8}+21q^{6}-37q^{4}-q^{2}+29-q^{-2}-37q^{-4}+21q^{-6}+36q^{-8}-53q^{-10}+54q^{-14}-37q^{-16}-18q^{-18}+37q^{-20}-10q^{-22}-14q^{-24}+11q^{-26}-3q^{-30}+q^{-32}$
3	$-q^{63}+3q^{61}-7q^{57}-2q^{55}+18q^{53}+13q^{51}-44q^{49}-36q^{47}+70q^{45}+93q^{43}-88q^{41}-184q^{39}+77q^{37}+291q^{35}-15q^{33}-384q^{31}-101q^{29}+441q^{27}+232q^{25}-426q^{23}-355q^{21}+345q^{19}+441q^{17}-229q^{15}-461q^{13}+93q^{11}+432q^{9}+41q^{7}-362q^{5}-164q^{3}+271q+271q^{-1}-164q^{-3}-362q^{-5}+41q^{-7}+432q^{-9}+93q^{-11}-461q^{-13}-229q^{-15}+441q^{-17}+345q^{-19}-355q^{-21}-426q^{-23}+232q^{-25}+441q^{-27}-101q^{-29}-384q^{-31}-15q^{-33}+291q^{-35}+77q^{-37}-184q^{-39}-88q^{-41}+93q^{-43}+70q^{-45}-36q^{-47}-44q^{-49}+13q^{-51}+18q^{-53}-2q^{-55}-7q^{-57}+3q^{-61}-q^{-63}$
4	$q^{104}-3q^{102}+7q^{98}-2q^{96}-2q^{94}-21q^{92}+4q^{90}+54q^{88}+13q^{86}-29q^{84}-144q^{82}-46q^{80}+234q^{78}+234q^{76}+24q^{74}-554q^{72}-518q^{70}+359q^{68}+966q^{66}+786q^{64}-922q^{62}-1844q^{60}-517q^{58}+1658q^{56}+2747q^{54}+35q^{52}-3124q^{50}-2894q^{48}+724q^{46}+4612q^{44}+2698q^{42}-2489q^{40}-5095q^{38}-2009q^{36}+4299q^{34}+5074q^{32}+92q^{30}-5038q^{28}-4396q^{26}+1926q^{24}+5220q^{22}+2525q^{20}-2979q^{18}-4789q^{16}-623q^{14}+3557q^{12}+3551q^{10}-574q^{8}-3759q^{6}-2380q^{4}+1504q^{2}+3699+1504q^{-2}-2380q^{-4}-3759q^{-6}-574q^{-8}+3551q^{-10}+3557q^{-12}-623q^{-14}-4789q^{-16}-2979q^{-18}+2525q^{-20}+5220q^{-22}+1926q^{-24}-4396q^{-26}-5038q^{-28}+92q^{-30}+5074q^{-32}+4299q^{-34}-2009q^{-36}-5095q^{-38}-2489q^{-40}+2698q^{-42}+4612q^{-44}+724q^{-46}-2894q^{-48}-3124q^{-50}+35q^{-52}+2747q^{-54}+1658q^{-56}-517q^{-58}-1844q^{-60}-922q^{-62}+786q^{-64}+966q^{-66}+359q^{-68}-518q^{-70}-554q^{-72}+24q^{-74}+234q^{-76}+234q^{-78}-46q^{-80}-144q^{-82}-29q^{-84}+13q^{-86}+54q^{-88}+4q^{-90}-21q^{-92}-2q^{-94}-2q^{-96}+7q^{-98}-3q^{-102}+q^{-104}$
5	$-q^{155}+3q^{153}-7q^{149}+2q^{147}+6q^{145}+5q^{143}+4q^{141}-14q^{139}-41q^{137}-10q^{135}+70q^{133}+105q^{131}+43q^{129}-135q^{127}-298q^{125}-225q^{123}+222q^{121}+732q^{119}+682q^{117}-159q^{115}-1354q^{113}-1787q^{111}-496q^{109}+2095q^{107}+3755q^{105}+2268q^{103}-2201q^{101}-6450q^{99}-5989q^{97}+635q^{95}+9175q^{93}+11773q^{91}+3865q^{89}-10226q^{87}-18885q^{85}-12188q^{83}+7671q^{81}+25415q^{79}+23813q^{77}-4q^{75}-28588q^{73}-36643q^{71}-13038q^{69}+26035q^{67}+47593q^{65}+29531q^{63}-16857q^{61}-53210q^{59}-46175q^{57}+1965q^{55}+51702q^{53}+59244q^{51}+15546q^{49}-43098q^{47}-65778q^{45}-32065q^{43}+29307q^{41}+65010q^{39}+44420q^{37}-13624q^{35}-57872q^{33}-50838q^{31}-1066q^{29}+46663q^{27}+51692q^{25}+12623q^{23}-34147q^{21}-48336q^{19}-20550q^{17}+22334q^{15}+42967q^{13}+25616q^{11}-12308q^{9}-37451q^{7}-29220q^{5}+3904q^{3}+32846q+32846q^{-1}+3904q^{-3}-29220q^{-5}-37451q^{-7}-12308q^{-9}+25616q^{-11}+42967q^{-13}+22334q^{-15}-20550q^{-17}-48336q^{-19}-34147q^{-21}+12623q^{-23}+51692q^{-25}+46663q^{-27}-1066q^{-29}-50838q^{-31}-57872q^{-33}-13624q^{-35}+44420q^{-37}+65010q^{-39}+29307q^{-41}-32065q^{-43}-65778q^{-45}-43098q^{-47}+15546q^{-49}+59244q^{-51}+51702q^{-53}+1965q^{-55}-46175q^{-57}-53210q^{-59}-16857q^{-61}+29531q^{-63}+47593q^{-65}+26035q^{-67}-13038q^{-69}-36643q^{-71}-28588q^{-73}-4q^{-75}+23813q^{-77}+25415q^{-79}+7671q^{-81}-12188q^{-83}-18885q^{-85}-10226q^{-87}+3865q^{-89}+11773q^{-91}+9175q^{-93}+635q^{-95}-5989q^{-97}-6450q^{-99}-2201q^{-101}+2268q^{-103}+3755q^{-105}+2095q^{-107}-496q^{-109}-1787q^{-111}-1354q^{-113}-159q^{-115}+682q^{-117}+732q^{-119}+222q^{-121}-225q^{-123}-298q^{-125}-135q^{-127}+43q^{-129}+105q^{-131}+70q^{-133}-10q^{-135}-41q^{-137}-14q^{-139}+4q^{-141}+5q^{-143}+6q^{-145}+2q^{-147}-7q^{-149}+3q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+2q^{12}-3q^{10}+3q^{8}-2q^{4}+3q^{2}-3+3q^{-2}-2q^{-4}+3q^{-8}-3q^{-10}+2q^{-12}+q^{-14}-q^{-16}$
2,0	$q^{42}-q^{40}-3q^{38}+q^{36}+7q^{34}+q^{32}-14q^{30}-3q^{28}+21q^{26}+4q^{24}-25q^{22}-5q^{20}+28q^{18}+8q^{16}-32q^{14}+q^{12}+26q^{10}-6q^{8}-15q^{6}+7q^{4}+7q^{2}-10+7q^{-2}+7q^{-4}-15q^{-6}-6q^{-8}+26q^{-10}+q^{-12}-32q^{-14}+8q^{-16}+28q^{-18}-5q^{-20}-25q^{-22}+4q^{-24}+21q^{-26}-3q^{-28}-14q^{-30}+q^{-32}+7q^{-34}+q^{-36}-3q^{-38}-q^{-40}+q^{-42}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+7q^{28}-14q^{26}+4q^{24}+22q^{22}-30q^{20}+5q^{18}+35q^{16}-41q^{14}+2q^{12}+35q^{10}-30q^{8}-6q^{6}+22q^{4}-2q^{2}-10-2q^{-2}+22q^{-4}-6q^{-6}-30q^{-8}+35q^{-10}+2q^{-12}-41q^{-14}+35q^{-16}+5q^{-18}-30q^{-20}+22q^{-22}+4q^{-24}-14q^{-26}+7q^{-28}+q^{-30}-3q^{-32}+q^{-34}$
1,0,0	$-q^{21}+q^{19}+2q^{15}-3q^{13}+4q^{11}-2q^{9}+2q^{7}-2q^{5}+2q^{3}-q-q^{-1}+2q^{-3}-2q^{-5}+2q^{-7}-2q^{-9}+4q^{-11}-3q^{-13}+2q^{-15}+q^{-19}-q^{-21}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3q^{32}-7q^{30}+13q^{28}-22q^{26}+32q^{24}-40q^{22}+48q^{20}-49q^{18}+45q^{16}-33q^{14}+16q^{12}+7q^{10}-32q^{8}+56q^{6}-76q^{4}+90q^{2}-96+90q^{-2}-76q^{-4}+56q^{-6}-32q^{-8}+7q^{-10}+16q^{-12}-33q^{-14}+45q^{-16}-49q^{-18}+48q^{-20}-40q^{-22}+32q^{-24}-22q^{-26}+13q^{-28}-7q^{-30}+3q^{-32}-q^{-34}$
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+10q^{46}-18q^{42}-13q^{40}+19q^{38}+31q^{36}-4q^{34}-43q^{32}-19q^{30}+40q^{28}+42q^{26}-21q^{24}-53q^{22}-4q^{20}+49q^{18}+22q^{16}-36q^{14}-31q^{12}+22q^{10}+33q^{8}-11q^{6}-33q^{4}+4q^{2}+35+4q^{-2}-33q^{-4}-11q^{-6}+33q^{-8}+22q^{-10}-31q^{-12}-36q^{-14}+22q^{-16}+49q^{-18}-4q^{-20}-53q^{-22}-21q^{-24}+42q^{-26}+40q^{-28}-19q^{-30}-43q^{-32}-4q^{-34}+31q^{-36}+19q^{-38}-13q^{-40}-18q^{-42}+10q^{-46}+4q^{-48}-3q^{-50}-3q^{-52}+q^{-56}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3q^{78}+7q^{76}-13q^{74}+15q^{72}-15q^{70}+4q^{68}+21q^{66}-53q^{64}+91q^{62}-111q^{60}+94q^{58}-33q^{56}-75q^{54}+204q^{52}-299q^{50}+314q^{48}-218q^{46}+18q^{44}+223q^{42}-417q^{40}+487q^{38}-379q^{36}+134q^{34}+154q^{32}-373q^{30}+418q^{28}-277q^{26}+21q^{24}+235q^{22}-365q^{20}+303q^{18}-66q^{16}-243q^{14}+490q^{12}-562q^{10}+415q^{8}-96q^{6}-286q^{4}+588q^{2}-699+588q^{-2}-286q^{-4}-96q^{-6}+415q^{-8}-562q^{-10}+490q^{-12}-243q^{-14}-66q^{-16}+303q^{-18}-365q^{-20}+235q^{-22}+21q^{-24}-277q^{-26}+418q^{-28}-373q^{-30}+154q^{-32}+134q^{-34}-379q^{-36}+487q^{-38}-417q^{-40}+223q^{-42}+18q^{-44}-218q^{-46}+314q^{-48}-299q^{-50}+204q^{-52}-75q^{-54}-33q^{-56}+94q^{-58}-111q^{-60}+91q^{-62}-53q^{-64}+21q^{-66}+4q^{-68}-15q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 88"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{3}+8t^{2}-24t+35-24t^{-1}+8t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}+2z^{4}-z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 101, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+13q^{2}-16q+17-16q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}-a^{4}z^{2}+2a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}-3z^{2}+a^{2}+a^{-2}-1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+6z^{8}a^{-2}+12z^{8}+7a^{3}z^{7}+14az^{7}+14z^{7}a^{-1}+7z^{7}a^{-3}+4a^{4}z^{6}-2a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}a^{-4}-12z^{6}+a^{5}z^{5}-11a^{3}z^{5}-32az^{5}-32z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-10a^{2}z^{4}-10z^{4}a^{-2}-6z^{4}a^{-4}-8z^{4}-a^{5}z^{3}+6a^{3}z^{3}+19az^{3}+19z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+3a^{4}z^{2}+7a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}+8z^{2}-a^{3}z-4az-4za^{-1}-za^{-3}-a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 88"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{3}+8t^{2}-24t+35-24t^{-1}+8t^{-2}-t^{-3}$, $-q^{5}+4q^{4}-8q^{3}+13q^{2}-16q+17-16q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $0$ $8$ $-{\frac {14}{3}}$ $-{\frac {34}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ ${\frac {136}{3}}$ ${\frac {449}{30}}$ $-{\frac {326}{5}}$ ${\frac {4978}{45}}$ $-{\frac {257}{18}}$ ${\frac {449}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 10 88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   3   3 7                 5 1   -4 5               8 3     5 3             8 5       -3 1           9 8         1 -1         8 9           1 -3       5 8             -3 -5     3 8               5 -7   1 5                 -4 -9   3                   3 -11 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{15}-4q^{14}+3q^{13}+12q^{12}-29q^{11}+7q^{10}+59q^{9}-84q^{8}-12q^{7}+150q^{6}-138q^{5}-65q^{4}+239q^{3}-153q^{2}-123q+275-123q^{-1}-153q^{-2}+239q^{-3}-65q^{-4}-138q^{-5}+150q^{-6}-12q^{-7}-84q^{-8}+59q^{-9}+7q^{-10}-29q^{-11}+12q^{-12}+3q^{-13}-4q^{-14}+q^{-15}$
3	$-q^{30}+4q^{29}-3q^{28}-7q^{27}+4q^{26}+24q^{25}-8q^{24}-64q^{23}+12q^{22}+130q^{21}+15q^{20}-245q^{19}-84q^{18}+391q^{17}+229q^{16}-551q^{15}-453q^{14}+674q^{13}+771q^{12}-760q^{11}-1111q^{10}+745q^{9}+1471q^{8}-664q^{7}-1781q^{6}+513q^{5}+2025q^{4}-325q^{3}-2172q^{2}+110q+2223+110q^{-1}-2172q^{-2}-325q^{-3}+2025q^{-4}+513q^{-5}-1781q^{-6}-664q^{-7}+1471q^{-8}+745q^{-9}-1111q^{-10}-760q^{-11}+771q^{-12}+674q^{-13}-453q^{-14}-551q^{-15}+229q^{-16}+391q^{-17}-84q^{-18}-245q^{-19}+15q^{-20}+130q^{-21}+12q^{-22}-64q^{-23}-8q^{-24}+24q^{-25}+4q^{-26}-7q^{-27}-3q^{-28}+4q^{-29}-q^{-30}$
4	$q^{50}-4q^{49}+3q^{48}+7q^{47}-9q^{46}+q^{45}-23q^{44}+28q^{43}+57q^{42}-50q^{41}-41q^{40}-138q^{39}+126q^{38}+337q^{37}-50q^{36}-251q^{35}-716q^{34}+162q^{33}+1214q^{32}+557q^{31}-431q^{30}-2424q^{29}-760q^{28}+2541q^{27}+2732q^{26}+658q^{25}-5136q^{24}-3919q^{23}+2771q^{22}+6350q^{21}+4546q^{20}-7050q^{19}-9106q^{18}+165q^{17}+9436q^{16}+10854q^{15}-6275q^{14}-14088q^{13}-4965q^{12}+10078q^{11}+17176q^{10}-2981q^{9}-16783q^{8}-10469q^{7}+8268q^{6}+21342q^{5}+1199q^{4}-16789q^{3}-14594q^{2}+5083q+22721+5083q^{-1}-14594q^{-2}-16789q^{-3}+1199q^{-4}+21342q^{-5}+8268q^{-6}-10469q^{-7}-16783q^{-8}-2981q^{-9}+17176q^{-10}+10078q^{-11}-4965q^{-12}-14088q^{-13}-6275q^{-14}+10854q^{-15}+9436q^{-16}+165q^{-17}-9106q^{-18}-7050q^{-19}+4546q^{-20}+6350q^{-21}+2771q^{-22}-3919q^{-23}-5136q^{-24}+658q^{-25}+2732q^{-26}+2541q^{-27}-760q^{-28}-2424q^{-29}-431q^{-30}+557q^{-31}+1214q^{-32}+162q^{-33}-716q^{-34}-251q^{-35}-50q^{-36}+337q^{-37}+126q^{-38}-138q^{-39}-41q^{-40}-50q^{-41}+57q^{-42}+28q^{-43}-23q^{-44}+q^{-45}-9q^{-46}+7q^{-47}+3q^{-48}-4q^{-49}+q^{-50}$
5	$-q^{75}+4q^{74}-3q^{73}-7q^{72}+9q^{71}+4q^{70}-2q^{69}+3q^{68}-21q^{67}-34q^{66}+40q^{65}+84q^{64}+33q^{63}-59q^{62}-199q^{61}-197q^{60}+113q^{59}+531q^{58}+543q^{57}-109q^{56}-1040q^{55}-1392q^{54}-320q^{53}+1822q^{52}+3134q^{51}+1551q^{50}-2527q^{49}-5861q^{48}-4569q^{47}+2283q^{46}+9758q^{45}+10091q^{44}+71q^{43}-13769q^{42}-18660q^{41}-6376q^{40}+16455q^{39}+29950q^{38}+17815q^{37}-15371q^{36}-42477q^{35}-34960q^{34}+8400q^{33}+53555q^{32}+56888q^{31}+6187q^{30}-60539q^{29}-81348q^{28}-27953q^{27}+60590q^{26}+105028q^{25}+55924q^{24}-52997q^{23}-125046q^{22}-86597q^{21}+37910q^{20}+138741q^{19}+117296q^{18}-17294q^{17}-145636q^{16}-144641q^{15}-6338q^{14}+145775q^{13}+167068q^{12}+30435q^{11}-140607q^{10}-183710q^{9}-53108q^{8}+131586q^{7}+194854q^{6}+73319q^{5}-119974q^{4}-201061q^{3}-91032q^{2}+106443q+203085+106443q^{-1}-91032q^{-2}-201061q^{-3}-119974q^{-4}+73319q^{-5}+194854q^{-6}+131586q^{-7}-53108q^{-8}-183710q^{-9}-140607q^{-10}+30435q^{-11}+167068q^{-12}+145775q^{-13}-6338q^{-14}-144641q^{-15}-145636q^{-16}-17294q^{-17}+117296q^{-18}+138741q^{-19}+37910q^{-20}-86597q^{-21}-125046q^{-22}-52997q^{-23}+55924q^{-24}+105028q^{-25}+60590q^{-26}-27953q^{-27}-81348q^{-28}-60539q^{-29}+6187q^{-30}+56888q^{-31}+53555q^{-32}+8400q^{-33}-34960q^{-34}-42477q^{-35}-15371q^{-36}+17815q^{-37}+29950q^{-38}+16455q^{-39}-6376q^{-40}-18660q^{-41}-13769q^{-42}+71q^{-43}+10091q^{-44}+9758q^{-45}+2283q^{-46}-4569q^{-47}-5861q^{-48}-2527q^{-49}+1551q^{-50}+3134q^{-51}+1822q^{-52}-320q^{-53}-1392q^{-54}-1040q^{-55}-109q^{-56}+543q^{-57}+531q^{-58}+113q^{-59}-197q^{-60}-199q^{-61}-59q^{-62}+33q^{-63}+84q^{-64}+40q^{-65}-34q^{-66}-21q^{-67}+3q^{-68}-2q^{-69}+4q^{-70}+9q^{-71}-7q^{-72}-3q^{-73}+4q^{-74}-q^{-75}$
6	$q^{105}-4q^{104}+3q^{103}+7q^{102}-9q^{101}-4q^{100}-3q^{99}+22q^{98}-10q^{97}-2q^{96}+44q^{95}-68q^{94}-58q^{93}-20q^{92}+145q^{91}+95q^{90}+46q^{89}+125q^{88}-408q^{87}-531q^{86}-317q^{85}+615q^{84}+967q^{83}+1024q^{82}+876q^{81}-1619q^{80}-3217q^{79}-3248q^{78}+420q^{77}+4033q^{76}+6961q^{75}+7385q^{74}-1329q^{73}-11203q^{72}-17531q^{71}-10354q^{70}+4368q^{69}+23577q^{68}+35863q^{67}+19355q^{66}-14959q^{65}-52800q^{64}-59173q^{63}-30996q^{62}+33402q^{61}+99926q^{60}+104116q^{59}+38455q^{58}-79161q^{57}-162940q^{56}-165555q^{55}-42543q^{54}+151254q^{53}+271312q^{52}+233769q^{51}+13377q^{50}-249520q^{49}-414572q^{48}-308643q^{47}+46760q^{46}+420480q^{45}+576578q^{44}+343984q^{43}-145228q^{42}-645910q^{41}-753595q^{40}-338824q^{39}+352823q^{38}+902403q^{37}+878082q^{36}+270412q^{35}-643372q^{34}-1182741q^{33}-942447q^{32}-39231q^{31}+986653q^{30}+1396470q^{29}+904710q^{28}-317470q^{27}-1371425q^{26}-1527787q^{25}-639520q^{24}+759338q^{23}+1682298q^{22}+1515191q^{21}+201994q^{20}-1266624q^{19}-1892397q^{18}-1214631q^{17}+352347q^{16}+1696603q^{15}+1921584q^{14}+704085q^{13}-991256q^{12}-2009256q^{11}-1615471q^{10}-55878q^{9}+1547088q^{8}+2110070q^{7}+1076094q^{6}-686020q^{5}-1969547q^{4}-1844533q^{3}-390299q^{2}+1337527q+2158285+1337527q^{-1}-390299q^{-2}-1844533q^{-3}-1969547q^{-4}-686020q^{-5}+1076094q^{-6}+2110070q^{-7}+1547088q^{-8}-55878q^{-9}-1615471q^{-10}-2009256q^{-11}-991256q^{-12}+704085q^{-13}+1921584q^{-14}+1696603q^{-15}+352347q^{-16}-1214631q^{-17}-1892397q^{-18}-1266624q^{-19}+201994q^{-20}+1515191q^{-21}+1682298q^{-22}+759338q^{-23}-639520q^{-24}-1527787q^{-25}-1371425q^{-26}-317470q^{-27}+904710q^{-28}+1396470q^{-29}+986653q^{-30}-39231q^{-31}-942447q^{-32}-1182741q^{-33}-643372q^{-34}+270412q^{-35}+878082q^{-36}+902403q^{-37}+352823q^{-38}-338824q^{-39}-753595q^{-40}-645910q^{-41}-145228q^{-42}+343984q^{-43}+576578q^{-44}+420480q^{-45}+46760q^{-46}-308643q^{-47}-414572q^{-48}-249520q^{-49}+13377q^{-50}+233769q^{-51}+271312q^{-52}+151254q^{-53}-42543q^{-54}-165555q^{-55}-162940q^{-56}-79161q^{-57}+38455q^{-58}+104116q^{-59}+99926q^{-60}+33402q^{-61}-30996q^{-62}-59173q^{-63}-52800q^{-64}-14959q^{-65}+19355q^{-66}+35863q^{-67}+23577q^{-68}+4368q^{-69}-10354q^{-70}-17531q^{-71}-11203q^{-72}-1329q^{-73}+7385q^{-74}+6961q^{-75}+4033q^{-76}+420q^{-77}-3248q^{-78}-3217q^{-79}-1619q^{-80}+876q^{-81}+1024q^{-82}+967q^{-83}+615q^{-84}-317q^{-85}-531q^{-86}-408q^{-87}+125q^{-88}+46q^{-89}+95q^{-90}+145q^{-91}-20q^{-92}-58q^{-93}-68q^{-94}+44q^{-95}-2q^{-96}-10q^{-97}+22q^{-98}-3q^{-99}-4q^{-100}-9q^{-101}+7q^{-102}+3q^{-103}-4q^{-104}+q^{-105}$
7	$-q^{140}+4q^{139}-3q^{138}-7q^{137}+9q^{136}+4q^{135}+3q^{134}-17q^{133}-15q^{132}+33q^{131}-8q^{130}-16q^{129}+42q^{128}+30q^{127}+13q^{126}-121q^{125}-168q^{124}+67q^{123}+72q^{122}+153q^{121}+345q^{120}+206q^{119}+28q^{118}-744q^{117}-1273q^{116}-608q^{115}+135q^{114}+1493q^{113}+2823q^{112}+2540q^{111}+1128q^{110}-2911q^{109}-7248q^{108}-7552q^{107}-4466q^{106}+3929q^{105}+14027q^{104}+18818q^{103}+16073q^{102}-110q^{101}-24199q^{100}-41450q^{99}-42981q^{98}-16792q^{97}+30828q^{96}+75952q^{95}+97722q^{94}+66617q^{93}-17353q^{92}-118190q^{91}-190921q^{90}-173528q^{89}-48185q^{88}+141342q^{87}+318844q^{86}+366831q^{85}+218101q^{84}-93530q^{83}-454180q^{82}-657775q^{81}-546567q^{80}-108698q^{79}+519225q^{78}+1018197q^{77}+1077733q^{76}+567378q^{75}-394097q^{74}-1360104q^{73}-1798579q^{72}-1366682q^{71}-78293q^{70}+1520419q^{69}+2618368q^{68}+2535184q^{67}+1042851q^{66}-1289355q^{65}-3349882q^{64}-3995819q^{63}-2580161q^{62}+454017q^{61}+3732829q^{60}+5554146q^{59}+4652596q^{58}+1132131q^{57}-3491647q^{56}-6921359q^{55}-7080887q^{54}-3483098q^{53}+2414596q^{52}+7768501q^{51}+9563059q^{50}+6458604q^{49}-417981q^{48}-7822357q^{47}-11746214q^{46}-9770005q^{45}-2404931q^{44}+6931124q^{43}+13296490q^{42}+13053332q^{41}+5814614q^{40}-5113233q^{39}-14002256q^{38}-15949372q^{37}-9459051q^{36}+2549624q^{35}+13795700q^{34}+18181824q^{33}+12981758q^{32}+470475q^{31}-12777028q^{30}-19616672q^{29}-16076808q^{28}-3614657q^{27}+11154661q^{26}+20259777q^{25}+18557545q^{24}+6592876q^{23}-9195152q^{22}-20238112q^{21}-20364308q^{20}-9198979q^{19}+7151404q^{18}+19740523q^{17}+21549728q^{16}+11338004q^{15}-5213041q^{14}-18966439q^{13}-22240226q^{12}-13018279q^{11}+3484032q^{10}+18080389q^{9}+22588382q^{8}+14321596q^{7}-1979218q^{6}-17180052q^{5}-22733021q^{4}-15371342q^{3}+639284q^{2}+16291800q+22770653+16291800q^{-1}+639284q^{-2}-15371342q^{-3}-22733021q^{-4}-17180052q^{-5}-1979218q^{-6}+14321596q^{-7}+22588382q^{-8}+18080389q^{-9}+3484032q^{-10}-13018279q^{-11}-22240226q^{-12}-18966439q^{-13}-5213041q^{-14}+11338004q^{-15}+21549728q^{-16}+19740523q^{-17}+7151404q^{-18}-9198979q^{-19}-20364308q^{-20}-20238112q^{-21}-9195152q^{-22}+6592876q^{-23}+18557545q^{-24}+20259777q^{-25}+11154661q^{-26}-3614657q^{-27}-16076808q^{-28}-19616672q^{-29}-12777028q^{-30}+470475q^{-31}+12981758q^{-32}+18181824q^{-33}+13795700q^{-34}+2549624q^{-35}-9459051q^{-36}-15949372q^{-37}-14002256q^{-38}-5113233q^{-39}+5814614q^{-40}+13053332q^{-41}+13296490q^{-42}+6931124q^{-43}-2404931q^{-44}-9770005q^{-45}-11746214q^{-46}-7822357q^{-47}-417981q^{-48}+6458604q^{-49}+9563059q^{-50}+7768501q^{-51}+2414596q^{-52}-3483098q^{-53}-7080887q^{-54}-6921359q^{-55}-3491647q^{-56}+1132131q^{-57}+4652596q^{-58}+5554146q^{-59}+3732829q^{-60}+454017q^{-61}-2580161q^{-62}-3995819q^{-63}-3349882q^{-64}-1289355q^{-65}+1042851q^{-66}+2535184q^{-67}+2618368q^{-68}+1520419q^{-69}-78293q^{-70}-1366682q^{-71}-1798579q^{-72}-1360104q^{-73}-394097q^{-74}+567378q^{-75}+1077733q^{-76}+1018197q^{-77}+519225q^{-78}-108698q^{-79}-546567q^{-80}-657775q^{-81}-454180q^{-82}-93530q^{-83}+218101q^{-84}+366831q^{-85}+318844q^{-86}+141342q^{-87}-48185q^{-88}-173528q^{-89}-190921q^{-90}-118190q^{-91}-17353q^{-92}+66617q^{-93}+97722q^{-94}+75952q^{-95}+30828q^{-96}-16792q^{-97}-42981q^{-98}-41450q^{-99}-24199q^{-100}-110q^{-101}+16073q^{-102}+18818q^{-103}+14027q^{-104}+3929q^{-105}-4466q^{-106}-7552q^{-107}-7248q^{-108}-2911q^{-109}+1128q^{-110}+2540q^{-111}+2823q^{-112}+1493q^{-113}+135q^{-114}-608q^{-115}-1273q^{-116}-744q^{-117}+28q^{-118}+206q^{-119}+345q^{-120}+153q^{-121}+72q^{-122}+67q^{-123}-168q^{-124}-121q^{-125}+13q^{-126}+30q^{-127}+42q^{-128}-16q^{-129}-8q^{-130}+33q^{-131}-15q^{-132}-17q^{-133}+3q^{-134}+4q^{-135}+9q^{-136}-7q^{-137}-3q^{-138}+4q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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