10 59

10_58

10_60

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 59's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 59 at Knotilus!

Two figure 8 knots on a loop, interlinked.

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code	1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 12 16
Conway Notation	[22,211,2]

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

[edit Notes on presentations of 10 59]

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 59"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 18 6 20 12 16

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,211,2]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 13.3899 A-Polynomial See Data:10 59/A-polynomial

[edit Notes for 10 59's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for 10 59's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}+a^{2}+4a^{-2}-3a^{-4}+a^{-6}-2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}+3az^{7}+8z^{7}a^{-1}+11z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-9z^{6}a^{-2}+z^{6}a^{-4}+5z^{6}a^{-6}-4z^{6}-9az^{5}-27z^{5}a^{-1}-28z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}-8z^{4}a^{-2}-11z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-6z^{4}+8az^{3}+21z^{3}a^{-1}+20z^{3}a^{-3}+4z^{3}a^{-5}-3z^{3}a^{-7}+3a^{2}z^{2}+11z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}a^{-8}+8z^{2}-2az-5za^{-1}-4za^{-3}+za^{-7}-a^{2}-4a^{-2}-3a^{-4}-a^{-6}-2$
The A2 invariant	$q^{10}-q^{6}+2q^{4}-2q^{2}+2q^{-2}-q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}-3q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{46}-2q^{44}+6q^{42}-10q^{40}+12q^{38}-10q^{36}-q^{34}+23q^{32}-44q^{30}+64q^{28}-63q^{26}+33q^{24}+18q^{22}-83q^{20}+135q^{18}-149q^{16}+112q^{14}-28q^{12}-76q^{10}+158q^{8}-183q^{6}+145q^{4}-55q^{2}-52+123q^{-2}-140q^{-4}+86q^{-6}+9q^{-8}-96q^{-10}+143q^{-12}-111q^{-14}+24q^{-16}+88q^{-18}-182q^{-20}+220q^{-22}-177q^{-24}+68q^{-26}+74q^{-28}-192q^{-30}+256q^{-32}-225q^{-34}+126q^{-36}+6q^{-38}-123q^{-40}+179q^{-42}-167q^{-44}+85q^{-46}+20q^{-48}-97q^{-50}+117q^{-52}-74q^{-54}-17q^{-56}+101q^{-58}-146q^{-60}+127q^{-62}-63q^{-64}-31q^{-66}+113q^{-68}-154q^{-70}+149q^{-72}-93q^{-74}+23q^{-76}+41q^{-78}-84q^{-80}+92q^{-82}-79q^{-84}+52q^{-86}-16q^{-88}-10q^{-90}+27q^{-92}-32q^{-94}+28q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_59.

A1 Invariants.

Weight	Invariant
1	$q^{7}-2q^{5}+3q^{3}-3q+3q^{-1}+2q^{-7}-4q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{22}-2q^{20}-2q^{18}+8q^{16}-3q^{14}-12q^{12}+17q^{10}+4q^{8}-26q^{6}+15q^{4}+17q^{2}-27+4q^{-2}+22q^{-4}-14q^{-6}-9q^{-8}+14q^{-10}+6q^{-12}-16q^{-14}-3q^{-16}+24q^{-18}-15q^{-20}-18q^{-22}+29q^{-24}-5q^{-26}-19q^{-28}+17q^{-30}+q^{-32}-9q^{-34}+5q^{-36}-2q^{-40}+q^{-42}$
3	$q^{45}-2q^{43}-2q^{41}+3q^{39}+8q^{37}-3q^{35}-19q^{33}-q^{31}+32q^{29}+16q^{27}-45q^{25}-44q^{23}+50q^{21}+79q^{19}-35q^{17}-115q^{15}+3q^{13}+142q^{11}+42q^{9}-147q^{7}-91q^{5}+129q^{3}+134q-102q^{-1}-154q^{-3}+61q^{-5}+165q^{-7}-19q^{-9}-156q^{-11}-18q^{-13}+137q^{-15}+54q^{-17}-106q^{-19}-87q^{-21}+62q^{-23}+119q^{-25}-17q^{-27}-137q^{-29}-42q^{-31}+145q^{-33}+91q^{-35}-128q^{-37}-134q^{-39}+103q^{-41}+152q^{-43}-64q^{-45}-143q^{-47}+23q^{-49}+121q^{-51}-q^{-53}-85q^{-55}-11q^{-57}+51q^{-59}+12q^{-61}-27q^{-63}-8q^{-65}+14q^{-67}+2q^{-69}-5q^{-71}+2q^{-75}-2q^{-79}+q^{-81}$
4	$q^{76}-2q^{74}-2q^{72}+3q^{70}+3q^{68}+8q^{66}-10q^{64}-19q^{62}-q^{60}+14q^{58}+53q^{56}-q^{54}-67q^{52}-64q^{50}-13q^{48}+157q^{46}+117q^{44}-58q^{42}-213q^{40}-229q^{38}+174q^{36}+366q^{34}+219q^{32}-227q^{30}-624q^{28}-168q^{26}+432q^{24}+724q^{22}+204q^{20}-792q^{18}-775q^{16}-15q^{14}+977q^{12}+909q^{10}-399q^{8}-1123q^{6}-743q^{4}+676q^{2}+1331+282q^{-2}-940q^{-4}-1202q^{-6}+105q^{-8}+1243q^{-10}+766q^{-12}-499q^{-14}-1218q^{-16}-332q^{-18}+871q^{-20}+921q^{-22}-84q^{-24}-976q^{-26}-609q^{-28}+410q^{-30}+928q^{-32}+334q^{-34}-601q^{-36}-855q^{-38}-183q^{-40}+800q^{-42}+823q^{-44}+3q^{-46}-971q^{-48}-901q^{-50}+374q^{-52}+1141q^{-54}+764q^{-56}-682q^{-58}-1373q^{-60}-296q^{-62}+941q^{-64}+1238q^{-66}-68q^{-68}-1213q^{-70}-735q^{-72}+345q^{-74}+1079q^{-76}+379q^{-78}-622q^{-80}-646q^{-82}-105q^{-84}+558q^{-86}+371q^{-88}-157q^{-90}-293q^{-92}-175q^{-94}+169q^{-96}+168q^{-98}-8q^{-100}-65q^{-102}-83q^{-104}+34q^{-106}+40q^{-108}-2q^{-110}-21q^{-114}+7q^{-116}+5q^{-118}-4q^{-120}+4q^{-122}-3q^{-124}+2q^{-126}-2q^{-130}+q^{-132}$
5	$q^{115}-2q^{113}-2q^{111}+3q^{109}+3q^{107}+3q^{105}+q^{103}-10q^{101}-19q^{99}-q^{97}+23q^{95}+35q^{93}+27q^{91}-23q^{89}-85q^{87}-89q^{85}+9q^{83}+138q^{81}+196q^{79}+96q^{77}-158q^{75}-378q^{73}-322q^{71}+72q^{69}+543q^{67}+695q^{65}+268q^{63}-568q^{61}-1177q^{59}-914q^{57}+266q^{55}+1540q^{53}+1828q^{51}+587q^{49}-1513q^{47}-2818q^{45}-1968q^{43}+793q^{41}+3448q^{39}+3691q^{37}+794q^{35}-3334q^{33}-5328q^{31}-3066q^{29}+2184q^{27}+6311q^{25}+5617q^{23}+15q^{21}-6259q^{19}-7859q^{17}-2875q^{15}+5033q^{13}+9235q^{11}+5829q^{9}-2791q^{7}-9491q^{5}-8329q^{3}+86q+8648q^{-1}+9869q^{-3}+2601q^{-5}-6974q^{-7}-10438q^{-9}-4742q^{-11}+4983q^{-13}+10058q^{-15}+6127q^{-17}-3010q^{-19}-9090q^{-21}-6793q^{-23}+1407q^{-25}+7849q^{-27}+6891q^{-29}-169q^{-31}-6607q^{-33}-6724q^{-35}-809q^{-37}+5491q^{-39}+6538q^{-41}+1725q^{-43}-4415q^{-45}-6478q^{-47}-2842q^{-49}+3226q^{-51}+6503q^{-53}+4267q^{-55}-1664q^{-57}-6445q^{-59}-5982q^{-61}-366q^{-63}+5954q^{-65}+7714q^{-67}+2969q^{-69}-4836q^{-71}-9102q^{-73}-5758q^{-75}+2879q^{-77}+9654q^{-79}+8445q^{-81}-279q^{-83}-9174q^{-85}-10342q^{-87}-2584q^{-89}+7509q^{-91}+11130q^{-93}+5151q^{-95}-5068q^{-97}-10600q^{-99}-6856q^{-101}+2367q^{-103}+8890q^{-105}+7465q^{-107}+29q^{-109}-6579q^{-111}-6955q^{-113}-1623q^{-115}+4154q^{-117}+5651q^{-119}+2375q^{-121}-2159q^{-123}-4050q^{-125}-2356q^{-127}+802q^{-129}+2556q^{-131}+1882q^{-133}-61q^{-135}-1413q^{-137}-1291q^{-139}-223q^{-141}+697q^{-143}+767q^{-145}+230q^{-147}-284q^{-149}-397q^{-151}-173q^{-153}+101q^{-155}+194q^{-157}+89q^{-159}-34q^{-161}-73q^{-163}-41q^{-165}+4q^{-167}+27q^{-169}+22q^{-171}-5q^{-173}-12q^{-175}-2q^{-179}-q^{-181}+5q^{-183}+q^{-185}-3q^{-187}+2q^{-189}-2q^{-193}+q^{-195}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}-q^{6}+2q^{4}-2q^{2}+2q^{-2}-q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}-3q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
2,0	$q^{28}-2q^{24}-2q^{22}+3q^{20}+5q^{18}-4q^{16}-6q^{14}+6q^{12}+9q^{10}-6q^{8}-11q^{6}+7q^{4}+12q^{2}-9-9q^{-2}+11q^{-4}+3q^{-6}-7q^{-8}+7q^{-12}-2q^{-14}+10q^{-18}-3q^{-20}-12q^{-22}+5q^{-24}+8q^{-26}-15q^{-28}-8q^{-30}+13q^{-32}+8q^{-34}-8q^{-36}-4q^{-38}+10q^{-40}+3q^{-42}-6q^{-44}-2q^{-46}+q^{-48}-q^{-52}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-2q^{18}+2q^{16}+3q^{14}-8q^{12}+7q^{10}+5q^{8}-16q^{6}+13q^{4}+6q^{2}-21+14q^{-2}+9q^{-4}-18q^{-6}+8q^{-8}+11q^{-10}-5q^{-12}-4q^{-14}+3q^{-16}+7q^{-18}-13q^{-20}-7q^{-22}+20q^{-24}-13q^{-26}-10q^{-28}+24q^{-30}-9q^{-32}-10q^{-34}+15q^{-36}-3q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$q^{13}+q^{9}-q^{7}+2q^{5}-3q^{3}+q-2q^{-1}+2q^{-3}+3q^{-7}+3q^{-9}+q^{-13}-3q^{-15}-4q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-2q^{18}+6q^{16}-9q^{14}+14q^{12}-19q^{10}+23q^{8}-26q^{6}+25q^{4}-22q^{2}+13-4q^{-2}-7q^{-4}+22q^{-6}-32q^{-8}+45q^{-10}-47q^{-12}+50q^{-14}-45q^{-16}+37q^{-18}-27q^{-20}+13q^{-22}-2q^{-24}-11q^{-26}+18q^{-28}-24q^{-30}+25q^{-32}-24q^{-34}+21q^{-36}-15q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{34}-2q^{30}-2q^{28}+4q^{26}+6q^{24}-3q^{22}-11q^{20}-3q^{18}+15q^{16}+13q^{14}-12q^{12}-22q^{10}+2q^{8}+27q^{6}+12q^{4}-23q^{2}-22+11q^{-2}+26q^{-4}+2q^{-6}-23q^{-8}-8q^{-10}+17q^{-12}+13q^{-14}-12q^{-16}-11q^{-18}+10q^{-20}+15q^{-22}-7q^{-24}-18q^{-26}+2q^{-28}+19q^{-30}+2q^{-32}-22q^{-34}-11q^{-36}+19q^{-38}+19q^{-40}-13q^{-42}-25q^{-44}+2q^{-46}+27q^{-48}+11q^{-50}-17q^{-52}-19q^{-54}+5q^{-56}+18q^{-58}+5q^{-60}-9q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-2q^{44}+6q^{42}-10q^{40}+12q^{38}-10q^{36}-q^{34}+23q^{32}-44q^{30}+64q^{28}-63q^{26}+33q^{24}+18q^{22}-83q^{20}+135q^{18}-149q^{16}+112q^{14}-28q^{12}-76q^{10}+158q^{8}-183q^{6}+145q^{4}-55q^{2}-52+123q^{-2}-140q^{-4}+86q^{-6}+9q^{-8}-96q^{-10}+143q^{-12}-111q^{-14}+24q^{-16}+88q^{-18}-182q^{-20}+220q^{-22}-177q^{-24}+68q^{-26}+74q^{-28}-192q^{-30}+256q^{-32}-225q^{-34}+126q^{-36}+6q^{-38}-123q^{-40}+179q^{-42}-167q^{-44}+85q^{-46}+20q^{-48}-97q^{-50}+117q^{-52}-74q^{-54}-17q^{-56}+101q^{-58}-146q^{-60}+127q^{-62}-63q^{-64}-31q^{-66}+113q^{-68}-154q^{-70}+149q^{-72}-93q^{-74}+23q^{-76}+41q^{-78}-84q^{-80}+92q^{-82}-79q^{-84}+52q^{-86}-16q^{-88}-10q^{-90}+27q^{-92}-32q^{-94}+28q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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 59"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}+a^{2}+4a^{-2}-3a^{-4}+a^{-6}-2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}+3az^{7}+8z^{7}a^{-1}+11z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-9z^{6}a^{-2}+z^{6}a^{-4}+5z^{6}a^{-6}-4z^{6}-9az^{5}-27z^{5}a^{-1}-28z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}-8z^{4}a^{-2}-11z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-6z^{4}+8az^{3}+21z^{3}a^{-1}+20z^{3}a^{-3}+4z^{3}a^{-5}-3z^{3}a^{-7}+3a^{2}z^{2}+11z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}a^{-8}+8z^{2}-2az-5za^{-1}-4za^{-3}+za^{-7}-a^{2}-4a^{-2}-3a^{-4}-a^{-6}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_40, K11n66,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_106,}

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 59"];

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]=

{ $t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$, $q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{9_40, K11n66,}

In[6]:=

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

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

Out[6]=

{10_106,}

Vassiliev invariants

V2 and V3:

(-1, -1)

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 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ ${\frac {38}{3}}$ $32$ ${\frac {112}{3}}$ ${\frac {64}{3}}$ $24$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ $-{\frac {152}{3}}$ ${\frac {4769}{30}}$ ${\frac {194}{5}}$ ${\frac {1738}{45}}$ ${\frac {319}{18}}$ $-{\frac {31}{30}}$

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=$2 is the signature of 10 59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   2   -2 11                 4 1   3 9               6 2     -4 7             6 4       2 5           6 6         0 3         6 6           0 1       4 7             3 -1     2 5               -3 -3   1 4                 3 -5   2                   -2 -7 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{20}-3q^{19}+2q^{18}+6q^{17}-17q^{16}+12q^{15}+22q^{14}-53q^{13}+26q^{12}+56q^{11}-100q^{10}+29q^{9}+95q^{8}-127q^{7}+16q^{6}+117q^{5}-119q^{4}-7q^{3}+112q^{2}-83q-25+81q^{-1}-39q^{-2}-27q^{-3}+40q^{-4}-9q^{-5}-14q^{-6}+11q^{-7}-3q^{-9}+q^{-10}$
3	$q^{39}-3q^{38}+2q^{37}+2q^{36}-q^{35}-8q^{34}+9q^{33}+14q^{32}-23q^{31}-27q^{30}+48q^{29}+53q^{28}-85q^{27}-101q^{26}+132q^{25}+175q^{24}-183q^{23}-267q^{22}+211q^{21}+391q^{20}-232q^{19}-504q^{18}+217q^{17}+610q^{16}-178q^{15}-691q^{14}+122q^{13}+730q^{12}-42q^{11}-748q^{10}-27q^{9}+711q^{8}+118q^{7}-665q^{6}-182q^{5}+573q^{4}+255q^{3}-481q^{2}-286q+358+307q^{-1}-245q^{-2}-291q^{-3}+138q^{-4}+251q^{-5}-56q^{-6}-191q^{-7}-q^{-8}+133q^{-9}+24q^{-10}-77q^{-11}-30q^{-12}+39q^{-13}+23q^{-14}-16q^{-15}-14q^{-16}+6q^{-17}+5q^{-18}-3q^{-20}+q^{-21}$
4	$q^{64}-3q^{63}+2q^{62}+2q^{61}-5q^{60}+8q^{59}-11q^{58}+11q^{57}+4q^{56}-33q^{55}+29q^{54}-13q^{53}+53q^{52}-2q^{51}-150q^{50}+47q^{49}+44q^{48}+229q^{47}-q^{46}-494q^{45}-71q^{44}+180q^{43}+757q^{42}+186q^{41}-1157q^{40}-612q^{39}+204q^{38}+1758q^{37}+886q^{36}-1891q^{35}-1692q^{34}-274q^{33}+2903q^{32}+2192q^{31}-2188q^{30}-2929q^{29}-1351q^{28}+3594q^{27}+3638q^{26}-1811q^{25}-3696q^{24}-2626q^{23}+3524q^{22}+4612q^{21}-991q^{20}-3719q^{19}-3609q^{18}+2852q^{17}+4866q^{16}-56q^{15}-3125q^{14}-4127q^{13}+1833q^{12}+4499q^{11}+836q^{10}-2120q^{9}-4177q^{8}+630q^{7}+3613q^{6}+1555q^{5}-855q^{4}-3700q^{3}-508q^{2}+2306q+1817+367q^{-1}-2651q^{-2}-1163q^{-3}+887q^{-4}+1437q^{-5}+1091q^{-6}-1343q^{-7}-1095q^{-8}-105q^{-9}+677q^{-10}+1074q^{-11}-347q^{-12}-575q^{-13}-397q^{-14}+77q^{-15}+618q^{-16}+50q^{-17}-129q^{-18}-250q^{-19}-115q^{-20}+215q^{-21}+66q^{-22}+26q^{-23}-75q^{-24}-75q^{-25}+45q^{-26}+15q^{-27}+23q^{-28}-9q^{-29}-21q^{-30}+6q^{-31}+5q^{-33}-3q^{-35}+q^{-36}$
5	$q^{95}-3q^{94}+2q^{93}+2q^{92}-5q^{91}+4q^{90}+5q^{89}-9q^{88}+q^{87}+4q^{86}-17q^{85}+11q^{84}+32q^{83}-4q^{82}-22q^{81}-41q^{80}-49q^{79}+50q^{78}+155q^{77}+101q^{76}-115q^{75}-315q^{74}-273q^{73}+163q^{72}+669q^{71}+638q^{70}-185q^{69}-1235q^{68}-1341q^{67}+41q^{66}+2021q^{65}+2581q^{64}+489q^{63}-2989q^{62}-4499q^{61}-1653q^{60}+3912q^{59}+7115q^{58}+3765q^{57}-4486q^{56}-10276q^{55}-6985q^{54}+4288q^{53}+13723q^{52}+11201q^{51}-3061q^{50}-16799q^{49}-16208q^{48}+544q^{47}+19255q^{46}+21420q^{45}+2918q^{44}-20420q^{43}-26301q^{42}-7214q^{41}+20423q^{40}+30315q^{39}+11642q^{38}-19211q^{37}-33076q^{36}-15851q^{35}+17079q^{34}+34581q^{33}+19447q^{32}-14466q^{31}-34836q^{30}-22171q^{29}+11463q^{28}+34118q^{27}+24228q^{26}-8535q^{25}-32600q^{24}-25448q^{23}+5395q^{22}+30482q^{21}+26291q^{20}-2395q^{19}-27787q^{18}-26495q^{17}-905q^{16}+24567q^{15}+26408q^{14}+4043q^{13}-20727q^{12}-25537q^{11}-7347q^{10}+16367q^{9}+24111q^{8}+10123q^{7}-11590q^{6}-21606q^{5}-12422q^{4}+6642q^{3}+18415q^{2}+13587q-2015-14338q^{-1}-13643q^{-2}-1920q^{-3}+10000q^{-4}+12425q^{-5}+4685q^{-6}-5718q^{-7}-10237q^{-8}-6122q^{-9}+2092q^{-10}+7441q^{-11}+6285q^{-12}+556q^{-13}-4635q^{-14}-5428q^{-15}-2035q^{-16}+2191q^{-17}+4023q^{-18}+2550q^{-19}-507q^{-20}-2531q^{-21}-2278q^{-22}-464q^{-23}+1262q^{-24}+1700q^{-25}+798q^{-26}-431q^{-27}-1037q^{-28}-752q^{-29}-12q^{-30}+520q^{-31}+535q^{-32}+178q^{-33}-201q^{-34}-325q^{-35}-164q^{-36}+49q^{-37}+141q^{-38}+122q^{-39}+19q^{-40}-71q^{-41}-64q^{-42}-9q^{-43}+12q^{-44}+24q^{-45}+23q^{-46}-9q^{-47}-14q^{-48}-q^{-49}+5q^{-52}-3q^{-54}+q^{-55}$
6	$q^{132}-3q^{131}+2q^{130}+2q^{129}-5q^{128}+4q^{127}+q^{126}+7q^{125}-19q^{124}+q^{123}+20q^{122}-25q^{121}+16q^{120}+18q^{119}+21q^{118}-73q^{117}-31q^{116}+64q^{115}-44q^{114}+82q^{113}+112q^{112}+42q^{111}-292q^{110}-247q^{109}+90q^{108}+21q^{107}+497q^{106}+606q^{105}+152q^{104}-1059q^{103}-1376q^{102}-425q^{101}+220q^{100}+2204q^{99}+2916q^{98}+1258q^{97}-2865q^{96}-5443q^{95}-3951q^{94}-726q^{93}+6525q^{92}+10723q^{91}+7429q^{90}-4175q^{89}-15120q^{88}-16463q^{87}-8829q^{86}+11686q^{85}+28398q^{84}+27511q^{83}+3240q^{82}-28283q^{81}-44153q^{80}-35885q^{79}+6963q^{78}+53013q^{77}+69038q^{76}+33878q^{75}-31299q^{74}-82485q^{73}-89733q^{72}-24463q^{71}+67566q^{70}+124650q^{69}+94827q^{68}-5378q^{67}-110736q^{66}-159124q^{65}-88087q^{64}+51617q^{63}+169514q^{62}+170878q^{61}+53259q^{60}-107320q^{59}-215770q^{58}-165370q^{57}+2969q^{56}+181264q^{55}+232043q^{54}+124295q^{53}-71352q^{52}-237847q^{51}-226639q^{50}-57865q^{49}+159895q^{48}+258830q^{47}+180526q^{46}-22299q^{45}-226671q^{44}-255929q^{43}-107466q^{42}+122916q^{41}+254329q^{40}+210290q^{39}+20838q^{38}-197919q^{37}-258084q^{36}-138340q^{35}+85001q^{34}+232503q^{33}+219810q^{32}+54199q^{31}-162683q^{30}-245235q^{29}-157373q^{28}+47338q^{27}+201336q^{26}+218957q^{25}+84517q^{24}-120303q^{23}-222151q^{22}-171612q^{21}+4097q^{20}+158404q^{19}+208727q^{18}+115433q^{17}-65976q^{16}-183941q^{15}-177651q^{14}-44691q^{13}+99222q^{12}+180816q^{11}+138738q^{10}-2953q^{9}-125398q^{8}-163229q^{7}-86074q^{6}+29317q^{5}+128622q^{4}+138288q^{3}+51737q^{2}-53665q-120386-100518q^{-1}-31211q^{-2}+60715q^{-3}+105510q^{-4}+76269q^{-5}+8646q^{-6}-59455q^{-7}-79642q^{-8}-59587q^{-9}+1761q^{-10}+53025q^{-11}+63614q^{-12}+38668q^{-13}-6747q^{-14}-38361q^{-15}-51131q^{-16}-26219q^{-17}+8138q^{-18}+31017q^{-19}+34299q^{-20}+17373q^{-21}-3803q^{-22}-24902q^{-23}-23742q^{-24}-11308q^{-25}+4420q^{-26}+15214q^{-27}+15812q^{-28}+9456q^{-29}-4507q^{-30}-9950q^{-31}-10160q^{-32}-5012q^{-33}+1515q^{-34}+6152q^{-35}+7515q^{-36}+2307q^{-37}-919q^{-38}-3667q^{-39}-3731q^{-40}-2189q^{-41}+469q^{-42}+2720q^{-43}+1729q^{-44}+1151q^{-45}-279q^{-46}-1025q^{-47}-1360q^{-48}-625q^{-49}+458q^{-50}+384q^{-51}+612q^{-52}+269q^{-53}-387q^{-55}-305q^{-56}+13q^{-57}-27q^{-58}+134q^{-59}+108q^{-60}+88q^{-61}-67q^{-62}-71q^{-63}+2q^{-64}-33q^{-65}+12q^{-66}+15q^{-67}+32q^{-68}-9q^{-69}-14q^{-70}+6q^{-71}-7q^{-72}+5q^{-75}-3q^{-77}+q^{-78}$
7	$q^{175}-3q^{174}+2q^{173}+2q^{172}-5q^{171}+4q^{170}+q^{169}+3q^{168}-3q^{167}-19q^{166}+17q^{165}+12q^{164}-20q^{163}+12q^{162}+3q^{161}+13q^{160}-18q^{159}-78q^{158}+56q^{157}+67q^{156}-14q^{155}+39q^{154}-35q^{153}-21q^{152}-111q^{151}-247q^{150}+152q^{149}+325q^{148}+266q^{147}+262q^{146}-232q^{145}-506q^{144}-785q^{143}-920q^{142}+363q^{141}+1475q^{140}+2037q^{139}+1839q^{138}-467q^{137}-2745q^{136}-4522q^{135}-4567q^{134}-245q^{133}+5220q^{132}+9665q^{131}+10101q^{130}+2561q^{129}-8323q^{128}-18334q^{127}-21216q^{126}-9528q^{125}+10914q^{124}+32221q^{123}+41491q^{122}+25211q^{121}-10127q^{120}-51064q^{119}-74442q^{118}-56212q^{117}-608q^{116}+72262q^{115}+123499q^{114}+110369q^{113}+30341q^{112}-89468q^{111}-187950q^{110}-194369q^{109}-91202q^{108}+90580q^{107}+262071q^{106}+312183q^{105}+195053q^{104}-60419q^{103}-333639q^{102}-459859q^{101}-349449q^{100}-17771q^{99}+383829q^{98}+625054q^{97}+554478q^{96}+156798q^{95}-391708q^{94}-787693q^{93}-798981q^{92}-360028q^{91}+339456q^{90}+921759q^{89}+1060754q^{88}+620370q^{87}-216558q^{86}-1004231q^{85}-1313097q^{84}-916642q^{83}+27372q^{82}+1017968q^{81}+1525433q^{80}+1221199q^{79}+214803q^{78}-959249q^{77}-1677609q^{76}-1503543q^{75}-482798q^{74}+836629q^{73}+1756491q^{72}+1737910q^{71}+750054q^{70}-668316q^{69}-1765174q^{68}-1909362q^{67}-990367q^{66}+478970q^{65}+1715130q^{64}+2013147q^{63}+1186804q^{62}-290062q^{61}-1624327q^{60}-2057064q^{59}-1333116q^{58}+119079q^{57}+1512667q^{56}+2053620q^{55}+1430994q^{54}+26504q^{53}-1393841q^{52}-2018872q^{51}-1491657q^{50}-146548q^{49}+1278259q^{48}+1966185q^{47}+1525924q^{46}+247682q^{45}-1165838q^{44}-1904384q^{43}-1547603q^{42}-339934q^{41}+1054432q^{40}+1836659q^{39}+1563270q^{38}+433405q^{37}-934402q^{36}-1760650q^{35}-1578328q^{34}-535662q^{33}+799050q^{32}+1669468q^{31}+1588648q^{30}+650127q^{29}-638978q^{28}-1554650q^{27}-1589201q^{26}-773601q^{25}+452900q^{24}+1406920q^{23}+1566532q^{22}+898023q^{21}-240689q^{20}-1220620q^{19}-1510509q^{18}-1009004q^{17}+13846q^{16}+994613q^{15}+1407490q^{14}+1089286q^{13}+214501q^{12}-735297q^{11}-1253458q^{10}-1122016q^{9}-421402q^{8}+458102q^{7}+1047594q^{6}+1093289q^{5}+586349q^{4}-183526q^{3}-802844q^{2}-999366q-688534-61652q^{-1}+538408q^{-2}+844839q^{-3}+717559q^{-4}+254425q^{-5}-281280q^{-6}-648043q^{-7}-673235q^{-8}-376793q^{-9}+58547q^{-10}+434006q^{-11}+568182q^{-12}+423484q^{-13}+108117q^{-14}-231142q^{-15}-425527q^{-16}-402092q^{-17}-207529q^{-18}+64525q^{-19}+272883q^{-20}+330682q^{-21}+241488q^{-22}+51003q^{-23}-135388q^{-24}-234540q^{-25}-223883q^{-26}-111853q^{-27}+31530q^{-28}+137589q^{-29}+174030q^{-30}+126180q^{-31}+32687q^{-32}-57695q^{-33}-113998q^{-34}-109320q^{-35}-59734q^{-36}+4273q^{-37}+59446q^{-38}+77364q^{-39}+60427q^{-40}+23421q^{-41}-19957q^{-42}-44848q^{-43}-46742q^{-44}-30772q^{-45}-2624q^{-46}+19304q^{-47}+29060q^{-48}+26577q^{-49}+11643q^{-50}-3586q^{-51}-14239q^{-52}-18098q^{-53}-12038q^{-54}-3445q^{-55}+4502q^{-56}+9813q^{-57}+8779q^{-58}+5232q^{-59}+428q^{-60}-4316q^{-61}-5140q^{-62}-4126q^{-63}-1847q^{-64}+1118q^{-65}+2202q^{-66}+2606q^{-67}+1934q^{-68}+94q^{-69}-822q^{-70}-1286q^{-71}-1165q^{-72}-354q^{-73}-12q^{-74}+464q^{-75}+739q^{-76}+340q^{-77}+92q^{-78}-184q^{-79}-290q^{-80}-119q^{-81}-142q^{-82}-34q^{-83}+142q^{-84}+103q^{-85}+74q^{-86}-9q^{-87}-56q^{-88}+6q^{-89}-33q^{-90}-33q^{-91}+12q^{-92}+15q^{-93}+23q^{-94}-14q^{-96}+6q^{-97}-7q^{-99}+5q^{-102}-3q^{-104}+q^{-105}$

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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