10 70

10_69

10_71

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10 70 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X5,16,6,17 X11,19,12,18 X13,1,14,20 X19,13,20,12 X17,15,18,14 X15,6,16,7
Gauss code	-1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 8, -7, 9, -10, 5, -9, 6, -8, 7
Dowker-Thistlethwaite code	4 8 16 10 2 18 20 6 14 12
Conway Notation	[22,3,2+]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 12.5109 A-Polynomial See Data:10 70/A-polynomial

[edit Notes for 10 70's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 70's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+16t-19+16t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 67, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+11q^{3}-11q^{2}+10q-8+5q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+4z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-5z^{2}+2a^{2}+3a^{-2}-2a^{-4}+a^{-6}-3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+2az^{7}+3z^{7}a^{-1}+6z^{7}a^{-3}+5z^{7}a^{-5}+a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}+5z^{6}a^{-6}-2z^{6}-6az^{5}-10z^{5}a^{-1}-11z^{5}a^{-3}-4z^{5}a^{-5}+3z^{5}a^{-7}-4a^{2}z^{4}-8z^{4}a^{-2}-12z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}-7z^{4}+5az^{3}+4z^{3}a^{-1}+2z^{3}a^{-3}-3z^{3}a^{-7}+5a^{2}z^{2}+9z^{2}a^{-2}+9z^{2}a^{-4}+4z^{2}a^{-6}-z^{2}a^{-8}+10z^{2}-az+za^{-3}+za^{-5}+za^{-7}-2a^{2}-3a^{-2}-2a^{-4}-a^{-6}-3$
The A2 invariant	$q^{10}+q^{8}+2q^{4}-2q^{2}-1+q^{-2}-2q^{-4}+3q^{-6}-q^{-8}+q^{-10}-2q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{46}-q^{44}+4q^{42}-5q^{40}+6q^{38}-5q^{36}+12q^{32}-23q^{30}+36q^{28}-37q^{26}+25q^{24}+5q^{22}-42q^{20}+80q^{18}-97q^{16}+85q^{14}-40q^{12}-32q^{10}+96q^{8}-132q^{6}+124q^{4}-70q^{2}-6+70q^{-2}-108q^{-4}+90q^{-6}-38q^{-8}-31q^{-10}+81q^{-12}-86q^{-14}+46q^{-16}+27q^{-18}-97q^{-20}+140q^{-22}-132q^{-24}+74q^{-26}+17q^{-28}-112q^{-30}+178q^{-32}-181q^{-34}+132q^{-36}-36q^{-38}-62q^{-40}+129q^{-42}-147q^{-44}+107q^{-46}-35q^{-48}-39q^{-50}+81q^{-52}-76q^{-54}+28q^{-56}+37q^{-58}-86q^{-60}+94q^{-62}-64q^{-64}+q^{-66}+61q^{-68}-107q^{-70}+119q^{-72}-89q^{-74}+41q^{-76}+16q^{-78}-61q^{-80}+80q^{-82}-76q^{-84}+57q^{-86}-25q^{-88}-3q^{-90}+23q^{-92}-32q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_70.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+3q^{3}-3q+2q^{-1}-q^{-3}+2q^{-7}-3q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{22}-q^{20}-q^{18}+5q^{16}-3q^{14}-8q^{12}+13q^{10}+q^{8}-19q^{6}+15q^{4}+11q^{2}-23+7q^{-2}+16q^{-4}-15q^{-6}-4q^{-8}+12q^{-10}+q^{-12}-12q^{-14}+q^{-16}+18q^{-18}-14q^{-20}-11q^{-22}+25q^{-24}-8q^{-26}-15q^{-28}+16q^{-30}-2q^{-32}-8q^{-34}+6q^{-36}-2q^{-40}+q^{-42}$
3	$q^{45}-q^{43}-q^{41}+q^{39}+4q^{37}-3q^{35}-9q^{33}+2q^{31}+19q^{29}+3q^{27}-30q^{25}-18q^{23}+42q^{21}+40q^{19}-41q^{17}-69q^{15}+28q^{13}+97q^{11}-6q^{9}-110q^{7}-28q^{5}+113q^{3}+60q-102q^{-1}-84q^{-3}+82q^{-5}+98q^{-7}-54q^{-9}-103q^{-11}+28q^{-13}+100q^{-15}-86q^{-19}-31q^{-21}+67q^{-23}+61q^{-25}-41q^{-27}-89q^{-29}+8q^{-31}+108q^{-33}+27q^{-35}-113q^{-37}-61q^{-39}+106q^{-41}+82q^{-43}-84q^{-45}-87q^{-47}+55q^{-49}+81q^{-51}-32q^{-53}-63q^{-55}+16q^{-57}+41q^{-59}-6q^{-61}-25q^{-63}+3q^{-65}+15q^{-67}-3q^{-69}-7q^{-71}+q^{-73}+3q^{-75}-2q^{-79}+q^{-81}$
4	$q^{76}-q^{74}-q^{72}+q^{70}+4q^{66}-5q^{64}-7q^{62}+4q^{60}+6q^{58}+21q^{56}-11q^{54}-36q^{52}-15q^{50}+14q^{48}+86q^{46}+26q^{44}-76q^{42}-110q^{40}-60q^{38}+172q^{36}+182q^{34}+3q^{32}-223q^{30}-308q^{28}+93q^{26}+359q^{24}+306q^{22}-122q^{20}-577q^{18}-250q^{16}+281q^{14}+626q^{12}+267q^{10}-557q^{8}-618q^{6}-95q^{4}+668q^{2}+660-241q^{-2}-716q^{-4}-472q^{-6}+434q^{-8}+789q^{-10}+107q^{-12}-569q^{-14}-632q^{-16}+156q^{-18}+686q^{-20}+315q^{-22}-348q^{-24}-609q^{-26}-79q^{-28}+480q^{-30}+452q^{-32}-90q^{-34}-511q^{-36}-340q^{-38}+182q^{-40}+565q^{-42}+267q^{-44}-296q^{-46}-611q^{-48}-251q^{-50}+536q^{-52}+630q^{-54}+92q^{-56}-682q^{-58}-678q^{-60}+251q^{-62}+738q^{-64}+489q^{-66}-434q^{-68}-794q^{-70}-100q^{-72}+485q^{-74}+591q^{-76}-79q^{-78}-545q^{-80}-233q^{-82}+144q^{-84}+391q^{-86}+80q^{-88}-229q^{-90}-138q^{-92}-15q^{-94}+157q^{-96}+57q^{-98}-68q^{-100}-34q^{-102}-24q^{-104}+48q^{-106}+14q^{-108}-23q^{-110}-8q^{-114}+14q^{-116}+2q^{-118}-8q^{-120}+2q^{-122}-2q^{-124}+3q^{-126}-2q^{-130}+q^{-132}$
5	$q^{115}-q^{113}-q^{111}+q^{109}+2q^{103}-3q^{101}-6q^{99}+4q^{97}+9q^{95}+6q^{93}+3q^{91}-16q^{89}-32q^{87}-12q^{85}+33q^{83}+62q^{81}+51q^{79}-23q^{77}-122q^{75}-139q^{73}-26q^{71}+167q^{69}+283q^{67}+174q^{65}-149q^{63}-450q^{61}-449q^{59}-31q^{57}+571q^{55}+826q^{53}+417q^{51}-472q^{49}-1195q^{47}-1057q^{45}+59q^{43}+1388q^{41}+1786q^{39}+737q^{37}-1158q^{35}-2424q^{33}-1840q^{31}+430q^{29}+2697q^{27}+2975q^{25}+785q^{23}-2366q^{21}-3903q^{19}-2282q^{17}+1468q^{15}+4314q^{13}+3714q^{11}-81q^{9}-4092q^{7}-4841q^{5}-1459q^{3}+3323q+5406q^{-1}+2871q^{-3}-2184q^{-5}-5405q^{-7}-3922q^{-9}+968q^{-11}+4944q^{-13}+4501q^{-15}+95q^{-17}-4211q^{-19}-4618q^{-21}-908q^{-23}+3416q^{-25}+4430q^{-27}+1417q^{-29}-2678q^{-31}-4082q^{-33}-1718q^{-35}+2030q^{-37}+3713q^{-39}+1961q^{-41}-1447q^{-43}-3398q^{-45}-2251q^{-47}+820q^{-49}+3094q^{-51}+2688q^{-53}-13q^{-55}-2745q^{-57}-3233q^{-59}-1017q^{-61}+2183q^{-63}+3765q^{-65}+2282q^{-67}-1304q^{-69}-4123q^{-71}-3622q^{-73}+86q^{-75}+4064q^{-77}+4820q^{-79}+1406q^{-81}-3501q^{-83}-5627q^{-85}-2893q^{-87}+2444q^{-89}+5778q^{-91}+4138q^{-93}-1068q^{-95}-5269q^{-97}-4847q^{-99}-306q^{-101}+4206q^{-103}+4863q^{-105}+1425q^{-107}-2857q^{-109}-4306q^{-111}-2051q^{-113}+1572q^{-115}+3346q^{-117}+2142q^{-119}-550q^{-121}-2285q^{-123}-1857q^{-125}-66q^{-127}+1372q^{-129}+1367q^{-131}+306q^{-133}-693q^{-135}-874q^{-137}-330q^{-139}+297q^{-141}+498q^{-143}+237q^{-145}-113q^{-147}-238q^{-149}-135q^{-151}+29q^{-153}+106q^{-155}+70q^{-157}-16q^{-159}-46q^{-161}-20q^{-163}+8q^{-165}+15q^{-167}+6q^{-169}-2q^{-171}-11q^{-173}-2q^{-175}+9q^{-177}+q^{-179}-3q^{-181}+q^{-183}-q^{-185}-2q^{-187}+3q^{-189}-2q^{-193}+q^{-195}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+2q^{16}+3q^{14}-4q^{12}+5q^{10}+5q^{8}-13q^{6}+8q^{4}+5q^{2}-19+10q^{-2}+8q^{-4}-16q^{-6}+6q^{-8}+11q^{-10}-6q^{-12}-2q^{-14}+4q^{-16}+5q^{-18}-9q^{-20}-6q^{-22}+16q^{-24}-10q^{-26}-10q^{-28}+20q^{-30}-6q^{-32}-10q^{-34}+14q^{-36}-2q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$q^{13}+q^{11}+2q^{9}+2q^{5}-3q^{3}-3q^{-1}+q^{-3}-q^{-5}+2q^{-7}+2q^{-9}+q^{-13}-2q^{-15}+q^{-17}-3q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+4q^{16}-5q^{14}+10q^{12}-13q^{10}+17q^{8}-19q^{6}+20q^{4}-19q^{2}+13-8q^{-2}-2q^{-4}+12q^{-6}-22q^{-8}+31q^{-10}-36q^{-12}+40q^{-14}-38q^{-16}+33q^{-18}-25q^{-20}+16q^{-22}-6q^{-24}-4q^{-26}+12q^{-28}-18q^{-30}+20q^{-32}-20q^{-34}+18q^{-36}-14q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{34}-q^{30}-q^{28}+3q^{26}+4q^{24}-q^{22}-7q^{20}-2q^{18}+10q^{16}+10q^{14}-8q^{12}-17q^{10}-q^{8}+20q^{6}+11q^{4}-17q^{2}-19+6q^{-2}+22q^{-4}+4q^{-6}-18q^{-8}-10q^{-10}+12q^{-12}+12q^{-14}-7q^{-16}-11q^{-18}+5q^{-20}+13q^{-22}-2q^{-24}-14q^{-26}-q^{-28}+15q^{-30}+5q^{-32}-15q^{-34}-11q^{-36}+13q^{-38}+16q^{-40}-8q^{-42}-21q^{-44}-q^{-46}+21q^{-48}+11q^{-50}-13q^{-52}-17q^{-54}+3q^{-56}+16q^{-58}+6q^{-60}-8q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+4q^{42}-5q^{40}+6q^{38}-5q^{36}+12q^{32}-23q^{30}+36q^{28}-37q^{26}+25q^{24}+5q^{22}-42q^{20}+80q^{18}-97q^{16}+85q^{14}-40q^{12}-32q^{10}+96q^{8}-132q^{6}+124q^{4}-70q^{2}-6+70q^{-2}-108q^{-4}+90q^{-6}-38q^{-8}-31q^{-10}+81q^{-12}-86q^{-14}+46q^{-16}+27q^{-18}-97q^{-20}+140q^{-22}-132q^{-24}+74q^{-26}+17q^{-28}-112q^{-30}+178q^{-32}-181q^{-34}+132q^{-36}-36q^{-38}-62q^{-40}+129q^{-42}-147q^{-44}+107q^{-46}-35q^{-48}-39q^{-50}+81q^{-52}-76q^{-54}+28q^{-56}+37q^{-58}-86q^{-60}+94q^{-62}-64q^{-64}+q^{-66}+61q^{-68}-107q^{-70}+119q^{-72}-89q^{-74}+41q^{-76}+16q^{-78}-61q^{-80}+80q^{-82}-76q^{-84}+57q^{-86}-25q^{-88}-3q^{-90}+23q^{-92}-32q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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 70"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+16t-19+16t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-3q^{6}+6q^{5}-9q^{4}+11q^{3}-11q^{2}+10q-8+5q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(-3, -2)

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 $-12$ $-16$ $72$ $82$ $46$ $192$ ${\frac {896}{3}}$ ${\frac {224}{3}}$ $80$ $-288$ $128$ $-984$ $-552$ $-{\frac {3871}{10}}$ ${\frac {2746}{15}}$ $-{\frac {3434}{5}}$ ${\frac {373}{2}}$ $-{\frac {1951}{10}}$

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 70. 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               5 2     -3 7             6 4       2 5           5 5         0 3         5 6           -1 1       4 6             2 -1     1 4               -3 -3   1 4                 3 -5   1                   -1 -7 1                     1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{20}-3q^{19}+2q^{18}+7q^{17}-17q^{16}+8q^{15}+25q^{14}-48q^{13}+15q^{12}+58q^{11}-84q^{10}+12q^{9}+90q^{8}-101q^{7}-q^{6}+103q^{5}-90q^{4}-17q^{3}+92q^{2}-59q-26+62q^{-1}-25q^{-2}-22q^{-3}+28q^{-4}-5q^{-5}-10q^{-6}+7q^{-7}-2q^{-9}+q^{-10}$
3	$q^{39}-3q^{38}+2q^{37}+3q^{36}-q^{35}-11q^{34}+6q^{33}+21q^{32}-13q^{31}-39q^{30}+25q^{29}+68q^{28}-38q^{27}-118q^{26}+56q^{25}+181q^{24}-64q^{23}-260q^{22}+59q^{21}+347q^{20}-40q^{19}-427q^{18}+7q^{17}+487q^{16}+41q^{15}-527q^{14}-90q^{13}+535q^{12}+143q^{11}-521q^{10}-188q^{9}+480q^{8}+229q^{7}-421q^{6}-260q^{5}+349q^{4}+278q^{3}-269q^{2}-276q+183+260q^{-1}-107q^{-2}-223q^{-3}+42q^{-4}+178q^{-5}-3q^{-6}-120q^{-7}-27q^{-8}+81q^{-9}+25q^{-10}-39q^{-11}-25q^{-12}+21q^{-13}+13q^{-14}-6q^{-15}-9q^{-16}+4q^{-17}+2q^{-18}-2q^{-20}+q^{-21}$
4	$q^{64}-3q^{63}+2q^{62}+3q^{61}-5q^{60}+5q^{59}-13q^{58}+12q^{57}+15q^{56}-27q^{55}+13q^{54}-36q^{53}+49q^{52}+49q^{51}-99q^{50}+3q^{49}-70q^{48}+174q^{47}+149q^{46}-271q^{45}-120q^{44}-161q^{43}+483q^{42}+460q^{41}-518q^{40}-497q^{39}-473q^{38}+949q^{37}+1130q^{36}-624q^{35}-1082q^{34}-1167q^{33}+1309q^{32}+2053q^{31}-375q^{30}-1569q^{29}-2096q^{28}+1305q^{27}+2827q^{26}+163q^{25}-1663q^{24}-2883q^{23}+945q^{22}+3142q^{21}+726q^{20}-1365q^{19}-3266q^{18}+423q^{17}+2971q^{16}+1147q^{15}-823q^{14}-3238q^{13}-136q^{12}+2441q^{11}+1408q^{10}-160q^{9}-2867q^{8}-666q^{7}+1653q^{6}+1471q^{5}+516q^{4}-2185q^{3}-1021q^{2}+747q+1227+991q^{-1}-1284q^{-2}-1013q^{-3}-16q^{-4}+704q^{-5}+1052q^{-6}-460q^{-7}-654q^{-8}-361q^{-9}+173q^{-10}+725q^{-11}-5q^{-12}-226q^{-13}-308q^{-14}-93q^{-15}+324q^{-16}+80q^{-17}-129q^{-19}-103q^{-20}+92q^{-21}+30q^{-22}+34q^{-23}-27q^{-24}-43q^{-25}+20q^{-26}+q^{-27}+13q^{-28}-2q^{-29}-11q^{-30}+5q^{-31}-q^{-32}+2q^{-33}-2q^{-35}+q^{-36}$
5	$q^{95}-3q^{94}+2q^{93}+3q^{92}-5q^{91}+q^{90}+3q^{89}-7q^{88}+6q^{87}+11q^{86}-16q^{85}-8q^{84}+12q^{83}+q^{82}+15q^{81}+4q^{80}-44q^{79}-34q^{78}+42q^{77}+87q^{76}+51q^{75}-73q^{74}-208q^{73}-137q^{72}+167q^{71}+437q^{70}+312q^{69}-274q^{68}-835q^{67}-681q^{66}+348q^{65}+1436q^{64}+1373q^{63}-269q^{62}-2273q^{61}-2472q^{60}-80q^{59}+3171q^{58}+4065q^{57}+935q^{56}-4047q^{55}-6095q^{54}-2335q^{53}+4620q^{52}+8347q^{51}+4373q^{50}-4704q^{49}-10607q^{48}-6876q^{47}+4198q^{46}+12548q^{45}+9579q^{44}-3064q^{43}-13941q^{42}-12213q^{41}+1464q^{40}+14674q^{39}+14486q^{38}+350q^{37}-14697q^{36}-16191q^{35}-2244q^{34}+14173q^{33}+17305q^{32}+3936q^{31}-13214q^{30}-17773q^{29}-5444q^{28}+11957q^{27}+17793q^{26}+6668q^{25}-10513q^{24}-17367q^{23}-7718q^{22}+8886q^{21}+16646q^{20}+8619q^{19}-7105q^{18}-15615q^{17}-9401q^{16}+5138q^{15}+14282q^{14}+10023q^{13}-3010q^{12}-12602q^{11}-10415q^{10}+814q^{9}+10572q^{8}+10430q^{7}+1296q^{6}-8196q^{5}-9972q^{4}-3162q^{3}+5682q^{2}+8947q+4517-3141q^{-1}-7437q^{-2}-5245q^{-3}+900q^{-4}+5565q^{-5}+5266q^{-6}+870q^{-7}-3642q^{-8}-4645q^{-9}-1946q^{-10}+1815q^{-11}+3645q^{-12}+2407q^{-13}-491q^{-14}-2455q^{-15}-2224q^{-16}-452q^{-17}+1375q^{-18}+1823q^{-19}+775q^{-20}-560q^{-21}-1175q^{-22}-850q^{-23}+46q^{-24}+707q^{-25}+637q^{-26}+163q^{-27}-286q^{-28}-441q^{-29}-209q^{-30}+105q^{-31}+219q^{-32}+162q^{-33}+15q^{-34}-118q^{-35}-100q^{-36}-11q^{-37}+26q^{-38}+49q^{-39}+32q^{-40}-19q^{-41}-26q^{-42}-3q^{-44}+4q^{-45}+12q^{-46}-3q^{-47}-7q^{-48}+3q^{-49}-q^{-51}+2q^{-52}-2q^{-54}+q^{-55}$
6	$q^{132}-3q^{131}+2q^{130}+3q^{129}-5q^{128}+q^{127}-q^{126}+9q^{125}-13q^{124}+2q^{123}+22q^{122}-27q^{121}-q^{120}+4q^{119}+34q^{118}-36q^{117}-15q^{116}+62q^{115}-76q^{114}+q^{113}+48q^{112}+120q^{111}-105q^{110}-118q^{109}+68q^{108}-209q^{107}+82q^{106}+303q^{105}+468q^{104}-208q^{103}-563q^{102}-329q^{101}-769q^{100}+296q^{99}+1335q^{98}+1900q^{97}+163q^{96}-1641q^{95}-2263q^{94}-3195q^{93}-80q^{92}+3845q^{91}+6496q^{90}+3249q^{89}-2433q^{88}-7081q^{87}-10749q^{86}-4297q^{85}+6688q^{84}+16430q^{83}+13649q^{82}+1459q^{81}-13393q^{80}-26300q^{79}-18239q^{78}+4228q^{77}+29736q^{76}+34569q^{75}+17057q^{74}-14246q^{73}-46801q^{72}-44892q^{71}-11370q^{70}+37819q^{69}+61374q^{68}+46633q^{67}-1246q^{66}-61813q^{65}-77627q^{64}-41330q^{63}+31612q^{62}+81939q^{61}+81732q^{60}+25819q^{59}-61850q^{58}-103039q^{57}-75846q^{56}+11423q^{55}+86852q^{54}+108599q^{53}+56676q^{52}-47652q^{51}-112329q^{50}-101819q^{49}-13005q^{48}+77413q^{47}+119771q^{46}+79786q^{45}-28125q^{44}-107350q^{43}-113575q^{42}-32309q^{41}+61603q^{40}+117779q^{39}+91486q^{38}-10585q^{37}-95095q^{36}-114274q^{35}-44687q^{34}+44956q^{33}+108708q^{32}+95399q^{31}+4470q^{30}-79576q^{29}-109084q^{28}-53794q^{27}+27115q^{26}+95237q^{25}+95588q^{24}+20213q^{23}-59806q^{22}-99283q^{21}-62145q^{20}+5585q^{19}+75646q^{18}+91588q^{17}+37316q^{16}-33836q^{15}-82040q^{14}-67029q^{13}-18577q^{12}+48061q^{11}+78980q^{10}+50761q^{9}-4104q^{8}-55164q^{7}-62105q^{6}-38341q^{5}+15871q^{4}+55016q^{3}+52520q^{2}+20731q-22807-44078q^{-1}-44550q^{-2}-11026q^{-3}+24632q^{-4}+39273q^{-5}+30852q^{-6}+4104q^{-7}-18706q^{-8}-34494q^{-9}-22650q^{-10}-415q^{-11}+17893q^{-12}+24460q^{-13}+15849q^{-14}+1816q^{-15}-16305q^{-16}-18376q^{-17}-11053q^{-18}+783q^{-19}+10573q^{-20}+13052q^{-21}+9531q^{-22}-2150q^{-23}-7705q^{-24}-9079q^{-25}-5370q^{-26}+390q^{-27}+5184q^{-28}+7230q^{-29}+2754q^{-30}-408q^{-31}-3408q^{-32}-3874q^{-33}-2504q^{-34}+243q^{-35}+2775q^{-36}+1961q^{-37}+1388q^{-38}-189q^{-39}-1166q^{-40}-1587q^{-41}-787q^{-42}+472q^{-43}+471q^{-44}+773q^{-45}+385q^{-46}-q^{-47}-493q^{-48}-402q^{-49}-5q^{-50}-54q^{-51}+185q^{-52}+169q^{-53}+125q^{-54}-92q^{-55}-103q^{-56}+7q^{-57}-62q^{-58}+14q^{-59}+30q^{-60}+52q^{-61}-18q^{-62}-22q^{-63}+17q^{-64}-17q^{-65}-2q^{-66}+14q^{-68}-4q^{-69}-8q^{-70}+7q^{-71}-2q^{-72}-q^{-74}+2q^{-75}-2q^{-77}+q^{-78}$
7	$q^{175}-3q^{174}+2q^{173}+3q^{172}-5q^{171}+q^{170}-q^{169}+5q^{168}+3q^{167}-17q^{166}+13q^{165}+11q^{164}-20q^{163}+q^{162}-4q^{161}+23q^{160}+12q^{159}-63q^{158}+29q^{157}+34q^{156}-34q^{155}+23q^{154}-18q^{153}+55q^{152}+12q^{151}-198q^{150}+3q^{149}+71q^{148}+47q^{147}+230q^{146}+50q^{145}+66q^{144}-145q^{143}-724q^{142}-353q^{141}+13q^{140}+520q^{139}+1332q^{138}+905q^{137}+355q^{136}-917q^{135}-2821q^{134}-2515q^{133}-1150q^{132}+1663q^{131}+5288q^{130}+5510q^{129}+3362q^{128}-1969q^{127}-9162q^{126}-11357q^{125}-8441q^{124}+1161q^{123}+14604q^{122}+21112q^{121}+18248q^{120}+2916q^{119}-20569q^{118}-35846q^{117}-35729q^{116}-13536q^{115}+25131q^{114}+55814q^{113}+63421q^{112}+34626q^{111}-24306q^{110}-79032q^{109}-102766q^{108}-71016q^{107}+12239q^{106}+101698q^{105}+153512q^{104}+125905q^{103}+16684q^{102}-117126q^{101}-211154q^{100}-199960q^{99}-68480q^{98}+117353q^{97}+269308q^{96}+290136q^{95}+144840q^{94}-95578q^{93}-318013q^{92}-388563q^{91}-243972q^{90}+46842q^{89}+348054q^{88}+485310q^{87}+358866q^{86}+28223q^{85}-352749q^{84}-568558q^{83}-478520q^{82}-124621q^{81}+328979q^{80}+629170q^{79}+591488q^{78}+232801q^{77}-279871q^{76}-661976q^{75}-686486q^{74}-341333q^{73}+212051q^{72}+666409q^{71}+756673q^{70}+440078q^{69}-134955q^{68}-647158q^{67}-799961q^{66}-520882q^{65}+58293q^{64}+610871q^{63}+817822q^{62}+580697q^{61}+11462q^{60}-565695q^{59}-816163q^{58}-619888q^{57}-69492q^{56}+517939q^{55}+800398q^{54}+642024q^{53}+116278q^{52}-471442q^{51}-777066q^{50}-652394q^{49}-153511q^{48}+427655q^{47}+749531q^{46}+655627q^{45}+185625q^{44}-384785q^{43}-719796q^{42}-655999q^{41}-216663q^{40}+340338q^{39}+687199q^{38}+654998q^{37}+250045q^{36}-290443q^{35}-649412q^{34}-652647q^{33}-287586q^{32}+232233q^{31}+603323q^{30}+646536q^{29}+328734q^{28}-164024q^{27}-545504q^{26}-632755q^{25}-370794q^{24}+86148q^{23}+473466q^{22}+606939q^{21}+408848q^{20}-1685q^{19}-386702q^{18}-564654q^{17}-436469q^{16}-83924q^{15}+287144q^{14}+503101q^{13}+447272q^{12}+163191q^{11}-180012q^{10}-422607q^{9}-435763q^{8}-227138q^{7}+73008q^{6}+326292q^{5}+399434q^{4}+268705q^{3}+24117q^{2}-221975q-339762-282160q^{-1}-101787q^{-2}+118926q^{-3}+262600q^{-4}+267060q^{-5}+152844q^{-6}-28173q^{-7}-177405q^{-8}-227350q^{-9}-174130q^{-10}-41158q^{-11}+95277q^{-12}+171190q^{-13}+167388q^{-14}+84278q^{-15}-26119q^{-16}-109769q^{-17}-139868q^{-18}-100408q^{-19}-22429q^{-20}+52993q^{-21}+100264q^{-22}+94696q^{-23}+49423q^{-24}-9197q^{-25}-59862q^{-26}-74724q^{-27}-55918q^{-28}-18180q^{-29}+25011q^{-30}+49184q^{-31}+49310q^{-32}+30175q^{-33}-1650q^{-34}-25698q^{-35}-35107q^{-36}-30072q^{-37}-11113q^{-38}+7919q^{-39}+20617q^{-40}+23757q^{-41}+14479q^{-42}+2152q^{-43}-8699q^{-44}-15163q^{-45}-12732q^{-46}-6330q^{-47}+1416q^{-48}+7932q^{-49}+8608q^{-50}+6383q^{-51}+2116q^{-52}-2954q^{-53}-4822q^{-54}-4734q^{-55}-2785q^{-56}+374q^{-57}+1933q^{-58}+2786q^{-59}+2423q^{-60}+587q^{-61}-550q^{-62}-1354q^{-63}-1434q^{-64}-631q^{-65}-213q^{-66}+446q^{-67}+864q^{-68}+490q^{-69}+218q^{-70}-137q^{-71}-324q^{-72}-175q^{-73}-247q^{-74}-81q^{-75}+171q^{-76}+125q^{-77}+112q^{-78}+4q^{-79}-47q^{-80}+22q^{-81}-62q^{-82}-62q^{-83}+19q^{-84}+18q^{-85}+32q^{-86}-4q^{-87}-19q^{-88}+25q^{-89}-4q^{-90}-16q^{-91}+10q^{-94}-2q^{-95}-9q^{-96}+6q^{-97}+2q^{-98}-2q^{-99}-q^{-101}+2q^{-102}-2q^{-104}+q^{-105}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 70]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[5, 16, 6, 17], X[11, 19, 12, 18], X[13, 1, 14, 20], X[19, 13, 20, 12], X[17, 15, 18, 14], X[15, 6, 16, 7]]
In[3]:=	GaussCode[Knot[10, 70]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 8, -7, 9, -10, 5, -9, 6, -8, 7]
In[4]:=	DTCode[Knot[10, 70]]
Out[4]=	DTCode[4, 8, 16, 10, 2, 18, 20, 6, 14, 12]
In[5]:=	br = BR[Knot[10, 70]]
Out[5]=	BR[5, {-1, 2, -1, -3, 2, 2, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[10, 70]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 70]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 70]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 70]][t]
Out[10]=	-3 7 16 2 3 -19 + t - -- + -- + 16 t - 7 t + t 2 t t
In[11]:=	Conway[Knot[10, 70]][z]
Out[11]=	2 4 6 1 - 3 z - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 70]}
In[13]:=	{KnotDet[Knot[10, 70]], KnotSignature[Knot[10, 70]]}
Out[13]=	{67, 2}
In[14]:=	Jones[Knot[10, 70]][q]
Out[14]=	-3 2 5 2 3 4 5 6 7 -8 + q - -- + - + 10 q - 11 q + 11 q - 9 q + 6 q - 3 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 70]}
In[16]:=	A2Invariant[Knot[10, 70]][q]
Out[16]=	-10 -8 2 2 2 4 6 8 10 14 -1 + q + q + -- - -- + q - 2 q + 3 q - q + q - 2 q + 4 2 q q 16 18 22 2 q - q + q
In[17]:=	HOMFLYPT[Knot[10, 70]][a, z]
Out[17]=	2 2 2 -6 2 3 2 2 z 4 z 4 z 2 2 4 -3 + a - -- + -- + 2 a - 5 z + -- - ---- + ---- + a z - 2 z - 4 2 6 4 2 a a a a a 4 4 6 2 z 3 z z ---- + ---- + -- 4 2 2 a a a
In[18]:=	Kauffman[Knot[10, 70]][a, z]
Out[18]=	2 2 -6 2 3 2 z z z 2 z 4 z -3 - a - -- - -- - 2 a + -- + -- + -- - a z + 10 z - -- + ---- + 4 2 7 5 3 8 6 a a a a a a a 2 2 3 3 3 4 9 z 9 z 2 2 3 z 2 z 4 z 3 4 z ---- + ---- + 5 a z - ---- + ---- + ---- + 5 a z - 7 z + -- - 4 2 7 3 a 8 a a a a a 4 4 4 5 5 5 5 6 z 12 z 8 z 2 4 3 z 4 z 11 z 10 z ---- - ----- - ---- - 4 a z + ---- - ---- - ----- - ----- - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 5 6 5 z 3 z 5 z 2 6 5 z 6 z 3 z 6 a z - 2 z + ---- + ---- - ---- + a z + ---- + ---- + ---- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 8 3 z 5 z z z 2 a z + 2 z + ---- + ---- + -- + -- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 70]], Vassiliev[3][Knot[10, 70]]}
Out[19]=	{-3, -2}
In[20]:=	Kh[Knot[10, 70]][q, t]
Out[20]=	3 1 1 1 4 1 4 4 q 6 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 6 q t + 5 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 11 4 11 5 13 5 15 6 4 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 70], 2][q]
Out[21]=	-10 2 7 10 5 28 22 25 62 2 -26 + q - -- + -- - -- - -- + -- - -- - -- + -- - 59 q + 92 q - 9 7 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 10 17 q - 90 q + 103 q - q - 101 q + 90 q + 12 q - 84 q + 11 12 13 14 15 16 17 18 58 q + 15 q - 48 q + 25 q + 8 q - 17 q + 7 q + 2 q - 19 20 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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