10 108

10_107

10_109

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10 108 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,4,17,3 X20,13,1,14 X14,7,15,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X12,6,13,5 X4,12,5,11 X2,16,3,15
Gauss code	1, -10, 2, -9, 8, -1, 4, -5, 6, -7, 9, -8, 3, -4, 10, -2, 7, -6, 5, -3
Dowker-Thistlethwaite code	6 16 12 14 18 4 20 2 10 8
Conway Notation	[30:20:20]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

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 [-4][-8] Hyperbolic Volume 12.9046 A-Polynomial See Data:10 108/A-polynomial

[edit Notes for 10 108'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 10 108's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+14t-15+14t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, 2 }
Jones polynomial	$-q^{6}+3q^{5}-5q^{4}+8q^{3}-10q^{2}+10q-9+8q^{-1}-5q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}-a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}+1$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+a^{3}z^{7}-3az^{7}+4z^{7}a^{-1}+8z^{7}a^{-3}-13a^{2}z^{6}-13z^{6}a^{-2}+7z^{6}a^{-4}-33z^{6}-4a^{3}z^{5}-11az^{5}-29z^{5}a^{-1}-17z^{5}a^{-3}+5z^{5}a^{-5}+17a^{2}z^{4}+4z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}+33z^{4}+5a^{3}z^{3}+19az^{3}+28z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-7a^{2}z^{2}+2z^{2}a^{-4}-z^{2}a^{-6}-10z^{2}-2a^{3}z-6az-6za^{-1}-2za^{-3}+1$
The A2 invariant	$-q^{12}+q^{10}+2q^{4}-q^{2}+2-q^{-4}+q^{-6}-2q^{-8}+2q^{-10}+q^{-16}-q^{-18}$
The G2 invariant	$q^{60}-2q^{58}+6q^{56}-11q^{54}+13q^{52}-12q^{50}-2q^{48}+25q^{46}-46q^{44}+58q^{42}-47q^{40}+11q^{38}+36q^{36}-81q^{34}+98q^{32}-77q^{30}+24q^{28}+37q^{26}-81q^{24}+89q^{22}-54q^{20}+3q^{18}+48q^{16}-68q^{14}+53q^{12}-11q^{10}-39q^{8}+74q^{6}-77q^{4}+59q^{2}-12-42q^{-2}+88q^{-4}-108q^{-6}+93q^{-8}-49q^{-10}-18q^{-12}+74q^{-14}-102q^{-16}+93q^{-18}-49q^{-20}-10q^{-22}+61q^{-24}-75q^{-26}+47q^{-28}-q^{-30}-45q^{-32}+67q^{-34}-50q^{-36}+12q^{-38}+31q^{-40}-57q^{-42}+63q^{-44}-46q^{-46}+16q^{-48}+13q^{-50}-36q^{-52}+41q^{-54}-36q^{-56}+27q^{-58}-12q^{-60}+2q^{-62}+9q^{-64}-21q^{-66}+24q^{-68}-23q^{-70}+16q^{-72}-7q^{-74}+7q^{-78}-12q^{-80}+13q^{-82}-10q^{-84}+7q^{-86}-2q^{-88}-q^{-90}+2q^{-92}-4q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

Further Quantum Invariants

Further quantum knot invariants for 10_108.

A1 Invariants.

Weight	Invariant
1	$-q^{9}+2q^{7}-2q^{5}+3q^{3}-q+q^{-1}-2q^{-5}+3q^{-7}-2q^{-9}+2q^{-11}-q^{-13}$
2	$q^{28}-2q^{26}-3q^{24}+7q^{22}+q^{20}-12q^{18}+7q^{16}+11q^{14}-14q^{12}-q^{10}+15q^{8}-8q^{6}-9q^{4}+12q^{2}+1-11q^{-2}+4q^{-4}+10q^{-6}-7q^{-8}-7q^{-10}+13q^{-12}+2q^{-14}-15q^{-16}+7q^{-18}+8q^{-20}-11q^{-22}+3q^{-24}+4q^{-26}-5q^{-28}+3q^{-30}-2q^{-34}+q^{-36}$
3	$-q^{57}+2q^{55}+3q^{53}-2q^{51}-10q^{49}-4q^{47}+18q^{45}+17q^{43}-16q^{41}-37q^{39}+q^{37}+50q^{35}+26q^{33}-46q^{31}-54q^{29}+24q^{27}+74q^{25}+7q^{23}-71q^{21}-42q^{19}+56q^{17}+64q^{15}-33q^{13}-74q^{11}+8q^{9}+74q^{7}+13q^{5}-70q^{3}-26q+68q^{-1}+38q^{-3}-58q^{-5}-51q^{-7}+50q^{-9}+62q^{-11}-30q^{-13}-73q^{-15}+69q^{-19}+36q^{-21}-51q^{-23}-69q^{-25}+25q^{-27}+82q^{-29}+10q^{-31}-76q^{-33}-33q^{-35}+56q^{-37}+39q^{-39}-31q^{-41}-31q^{-43}+12q^{-45}+17q^{-47}-3q^{-49}-8q^{-51}+2q^{-53}+2q^{-57}-q^{-61}-q^{-63}+2q^{-67}-q^{-69}$
4	$q^{96}-2q^{94}-3q^{92}+2q^{90}+5q^{88}+13q^{86}-3q^{84}-23q^{82}-24q^{80}-9q^{78}+55q^{76}+60q^{74}+7q^{72}-71q^{70}-131q^{68}-17q^{66}+112q^{64}+179q^{62}+81q^{60}-177q^{58}-240q^{56}-119q^{54}+180q^{52}+358q^{50}+134q^{48}-188q^{46}-425q^{44}-215q^{42}+262q^{40}+454q^{38}+265q^{36}-288q^{34}-548q^{32}-222q^{30}+304q^{28}+596q^{26}+194q^{24}-407q^{22}-555q^{20}-131q^{18}+493q^{16}+510q^{14}-40q^{12}-514q^{10}-390q^{8}+218q^{6}+505q^{4}+167q^{2}-353-402q^{-2}+79q^{-4}+433q^{-6}+199q^{-8}-302q^{-10}-397q^{-12}+18q^{-14}+442q^{-16}+306q^{-18}-215q^{-20}-473q^{-22}-228q^{-24}+326q^{-26}+519q^{-28}+156q^{-30}-357q^{-32}-592q^{-34}-154q^{-36}+466q^{-38}+620q^{-40}+147q^{-42}-592q^{-44}-648q^{-46}-q^{-48}+631q^{-50}+591q^{-52}-165q^{-54}-627q^{-56}-363q^{-58}+240q^{-60}+520q^{-62}+138q^{-64}-266q^{-66}-296q^{-68}-22q^{-70}+225q^{-72}+117q^{-74}-49q^{-76}-106q^{-78}-45q^{-80}+62q^{-82}+30q^{-84}-q^{-86}-18q^{-88}-18q^{-90}+19q^{-92}+q^{-94}-2q^{-98}-6q^{-100}+6q^{-102}-q^{-104}+q^{-106}-2q^{-110}+q^{-112}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{12}+q^{10}+2q^{4}-q^{2}+2-q^{-4}+q^{-6}-2q^{-8}+2q^{-10}+q^{-16}-q^{-18}$
1,1	$q^{36}-4q^{34}+14q^{32}-38q^{30}+74q^{28}-132q^{26}+196q^{24}-256q^{22}+308q^{20}-318q^{18}+286q^{16}-204q^{14}+95q^{12}+44q^{10}-196q^{8}+322q^{6}-425q^{4}+492q^{2}-524+502q^{-2}-427q^{-4}+330q^{-6}-200q^{-8}+74q^{-10}+48q^{-12}-134q^{-14}+180q^{-16}-198q^{-18}+178q^{-20}-152q^{-22}+130q^{-24}-104q^{-26}+83q^{-28}-70q^{-30}+70q^{-32}-64q^{-34}+48q^{-36}-42q^{-38}+38q^{-40}-28q^{-42}+18q^{-44}-12q^{-46}+8q^{-48}-4q^{-50}+q^{-52}$
2,0	$q^{34}-q^{32}-3q^{30}+3q^{26}+q^{24}-4q^{22}-q^{20}+7q^{18}+3q^{16}-5q^{14}+4q^{10}+3q^{8}-6q^{6}-2q^{4}+4q^{2}-3-q^{-2}+2q^{-4}-2q^{-8}+5q^{-10}+3q^{-12}-4q^{-14}-q^{-16}+7q^{-18}+3q^{-20}-10q^{-22}+q^{-24}+6q^{-26}-2q^{-28}-4q^{-30}+3q^{-34}-q^{-38}+q^{-40}-q^{-42}-q^{-44}+q^{-46}$

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{32}-q^{30}+3q^{26}-2q^{24}-2q^{22}+4q^{20}-2q^{18}-7q^{16}+3q^{14}+3q^{12}-6q^{10}-3q^{8}+12q^{6}+5q^{4}-9q^{2}+6+15q^{-2}-7q^{-4}-7q^{-6}+11q^{-8}-q^{-10}-9q^{-12}+2q^{-14}+5q^{-16}-6q^{-18}-5q^{-20}+8q^{-22}+q^{-24}-9q^{-26}+3q^{-28}+9q^{-30}-5q^{-32}-4q^{-34}+5q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+2q^{-44}+2q^{-46}-2q^{-48}-q^{-50}+q^{-52}$
1,0,0,0	$-q^{18}+q^{16}-q^{14}+q^{12}+q^{10}-q^{8}+2q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-18}+q^{-20}+q^{-22}-q^{-24}+q^{-26}-q^{-28}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{26}+2q^{24}-6q^{22}+9q^{20}-12q^{18}+16q^{16}-16q^{14}+18q^{12}-15q^{10}+12q^{8}-5q^{6}-2q^{4}+11q^{2}-20+26q^{-2}-31q^{-4}+32q^{-6}-31q^{-8}+27q^{-10}-20q^{-12}+14q^{-14}-5q^{-16}-2q^{-18}+9q^{-20}-13q^{-22}+15q^{-24}-15q^{-26}+14q^{-28}-12q^{-30}+9q^{-32}-7q^{-34}+5q^{-36}-3q^{-38}+2q^{-40}-q^{-42}$
1,0	$q^{44}-2q^{40}-2q^{38}+4q^{36}+5q^{34}-5q^{32}-9q^{30}+q^{28}+13q^{26}+4q^{24}-14q^{22}-11q^{20}+11q^{18}+17q^{16}-2q^{14}-18q^{12}-5q^{10}+15q^{8}+13q^{6}-8q^{4}-14q^{2}+2+12q^{-2}+2q^{-4}-10q^{-6}-3q^{-8}+10q^{-10}+5q^{-12}-10q^{-14}-7q^{-16}+10q^{-18}+12q^{-20}-6q^{-22}-15q^{-24}+q^{-26}+15q^{-28}+4q^{-30}-14q^{-32}-10q^{-34}+8q^{-36}+14q^{-38}-q^{-40}-12q^{-42}-6q^{-44}+7q^{-46}+10q^{-48}-7q^{-52}-5q^{-54}+2q^{-56}+5q^{-58}+q^{-60}-2q^{-62}-2q^{-64}+q^{-68}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{34}-2q^{32}+4q^{30}-6q^{28}+9q^{26}-11q^{24}+11q^{22}-14q^{20}+15q^{18}-15q^{16}+11q^{14}-10q^{12}+9q^{10}-3q^{8}-q^{6}+7q^{4}-8q^{2}+18-18q^{-2}+23q^{-4}-23q^{-6}+26q^{-8}-25q^{-10}+23q^{-12}-21q^{-14}+16q^{-16}-14q^{-18}+8q^{-20}-4q^{-22}-q^{-24}+5q^{-26}-8q^{-28}+10q^{-30}-10q^{-32}+13q^{-34}-12q^{-36}+9q^{-38}-9q^{-40}+10q^{-42}-7q^{-44}+4q^{-46}-5q^{-48}+5q^{-50}-2q^{-52}+q^{-54}-2q^{-56}+q^{-58}$

G2 Invariants.

Weight

Invariant

1,0

$q^{60}-2q^{58}+6q^{56}-11q^{54}+13q^{52}-12q^{50}-2q^{48}+25q^{46}-46q^{44}+58q^{42}-47q^{40}+11q^{38}+36q^{36}-81q^{34}+98q^{32}-77q^{30}+24q^{28}+37q^{26}-81q^{24}+89q^{22}-54q^{20}+3q^{18}+48q^{16}-68q^{14}+53q^{12}-11q^{10}-39q^{8}+74q^{6}-77q^{4}+59q^{2}-12-42q^{-2}+88q^{-4}-108q^{-6}+93q^{-8}-49q^{-10}-18q^{-12}+74q^{-14}-102q^{-16}+93q^{-18}-49q^{-20}-10q^{-22}+61q^{-24}-75q^{-26}+47q^{-28}-q^{-30}-45q^{-32}+67q^{-34}-50q^{-36}+12q^{-38}+31q^{-40}-57q^{-42}+63q^{-44}-46q^{-46}+16q^{-48}+13q^{-50}-36q^{-52}+41q^{-54}-36q^{-56}+27q^{-58}-12q^{-60}+2q^{-62}+9q^{-64}-21q^{-66}+24q^{-68}-23q^{-70}+16q^{-72}-7q^{-74}+7q^{-78}-12q^{-80}+13q^{-82}-10q^{-84}+7q^{-86}-2q^{-88}-q^{-90}+2q^{-92}-4q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

.

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["10 108"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+14t-15+14t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 63, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+a^{3}z^{7}-3az^{7}+4z^{7}a^{-1}+8z^{7}a^{-3}-13a^{2}z^{6}-13z^{6}a^{-2}+7z^{6}a^{-4}-33z^{6}-4a^{3}z^{5}-11az^{5}-29z^{5}a^{-1}-17z^{5}a^{-3}+5z^{5}a^{-5}+17a^{2}z^{4}+4z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}+33z^{4}+5a^{3}z^{3}+19az^{3}+28z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-7a^{2}z^{2}+2z^{2}a^{-4}-z^{2}a^{-6}-10z^{2}-2a^{3}z-6az-6za^{-1}-2za^{-3}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n161, ...}

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

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $0$ $0$ $-32$ $-32$ $0$ $0$ $-64$ $64$ $0$ $0$ $0$ $0$ $-16$ $-{\frac {352}{3}}$ ${\frac {512}{3}}$ $-{\frac {208}{3}}$ $16$

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 10 108. 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 13                     1 -1 11                   2   2 9                 3 1   -2 7               5 2     3 5             5 3       -2 3           5 5         0 1         5 6           1 -1       3 4             -1 -3     2 5               3 -5   1 3                 -2 -7   2                   2 -9 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{17}-3q^{16}+2q^{15}+4q^{14}-11q^{13}+11q^{12}+3q^{11}-25q^{10}+30q^{9}+2q^{8}-47q^{7}+47q^{6}+13q^{5}-67q^{4}+47q^{3}+30q^{2}-73q+32+42q^{-1}-62q^{-2}+11q^{-3}+43q^{-4}-39q^{-5}-5q^{-6}+30q^{-7}-14q^{-8}-9q^{-9}+11q^{-10}-q^{-11}-3q^{-12}+q^{-13}$
3	$-q^{33}+3q^{32}-2q^{31}-q^{30}-q^{29}+4q^{28}-3q^{26}+q^{25}-6q^{24}+5q^{23}+17q^{22}-4q^{21}-49q^{20}+5q^{19}+87q^{18}+13q^{17}-138q^{16}-38q^{15}+173q^{14}+85q^{13}-195q^{12}-132q^{11}+191q^{10}+172q^{9}-162q^{8}-201q^{7}+118q^{6}+215q^{5}-70q^{4}-213q^{3}+17q^{2}+208q+26-183q^{-1}-77q^{-2}+164q^{-3}+109q^{-4}-122q^{-5}-143q^{-6}+82q^{-7}+150q^{-8}-25q^{-9}-151q^{-10}-16q^{-11}+121q^{-12}+53q^{-13}-84q^{-14}-66q^{-15}+43q^{-16}+61q^{-17}-12q^{-18}-42q^{-19}-6q^{-20}+23q^{-21}+9q^{-22}-9q^{-23}-5q^{-24}+q^{-25}+3q^{-26}-q^{-27}$
4	$q^{54}-3q^{53}+2q^{52}+q^{51}-2q^{50}+8q^{49}-15q^{48}+6q^{47}+3q^{46}-q^{45}+26q^{44}-52q^{43}+6q^{42}+20q^{41}+30q^{40}+58q^{39}-159q^{38}-55q^{37}+77q^{36}+196q^{35}+166q^{34}-406q^{33}-329q^{32}+107q^{31}+600q^{30}+548q^{29}-686q^{28}-932q^{27}-157q^{26}+1062q^{25}+1304q^{24}-646q^{23}-1564q^{22}-804q^{21}+1118q^{20}+2043q^{19}-173q^{18}-1718q^{17}-1424q^{16}+680q^{15}+2278q^{14}+340q^{13}-1355q^{12}-1617q^{11}+126q^{10}+2033q^{9}+598q^{8}-834q^{7}-1481q^{6}-298q^{5}+1618q^{4}+693q^{3}-333q^{2}-1247q-652+1137q^{-1}+742q^{-2}+187q^{-3}-909q^{-4}-939q^{-5}+529q^{-6}+618q^{-7}+661q^{-8}-359q^{-9}-956q^{-10}-95q^{-11}+194q^{-12}+809q^{-13}+242q^{-14}-554q^{-15}-387q^{-16}-332q^{-17}+483q^{-18}+502q^{-19}-q^{-20}-198q^{-21}-524q^{-22}+6q^{-23}+292q^{-24}+236q^{-25}+124q^{-26}-300q^{-27}-172q^{-28}-7q^{-29}+115q^{-30}+187q^{-31}-42q^{-32}-74q^{-33}-74q^{-34}-14q^{-35}+73q^{-36}+18q^{-37}+4q^{-38}-21q^{-39}-19q^{-40}+9q^{-41}+3q^{-42}+5q^{-43}-q^{-44}-3q^{-45}+q^{-46}$
5	$-q^{80}+3q^{79}-2q^{78}-q^{77}+2q^{76}-5q^{75}+3q^{74}+9q^{73}-6q^{72}-9q^{71}+5q^{70}-3q^{69}+12q^{68}+15q^{67}-28q^{66}-37q^{65}+13q^{64}+65q^{63}+65q^{62}-26q^{61}-158q^{60}-168q^{59}+71q^{58}+378q^{57}+350q^{56}-152q^{55}-729q^{54}-724q^{53}+169q^{52}+1349q^{51}+1442q^{50}-140q^{49}-2183q^{48}-2577q^{47}-246q^{46}+3200q^{45}+4288q^{44}+1094q^{43}-4137q^{42}-6440q^{41}-2673q^{40}+4693q^{39}+8806q^{38}+4912q^{37}-4481q^{36}-10951q^{35}-7654q^{34}+3439q^{33}+12407q^{32}+10374q^{31}-1604q^{30}-12858q^{29}-12675q^{28}-612q^{27}+12362q^{26}+14075q^{25}+2765q^{24}-11063q^{23}-14553q^{22}-4520q^{21}+9456q^{20}+14225q^{19}+5636q^{18}-7799q^{17}-13395q^{16}-6263q^{15}+6350q^{14}+12398q^{13}+6554q^{12}-5111q^{11}-11423q^{10}-6748q^{9}+3968q^{8}+10498q^{7}+7043q^{6}-2763q^{5}-9620q^{4}-7381q^{3}+1392q^{2}+8515q+7784+201q^{-1}-7191q^{-2}-7951q^{-3}-1886q^{-4}+5414q^{-5}+7779q^{-6}+3519q^{-7}-3373q^{-8}-7002q^{-9}-4775q^{-10}+1077q^{-11}+5663q^{-12}+5432q^{-13}+1027q^{-14}-3732q^{-15}-5278q^{-16}-2755q^{-17}+1614q^{-18}+4338q^{-19}+3624q^{-20}+437q^{-21}-2757q^{-22}-3680q^{-23}-1900q^{-24}+1000q^{-25}+2843q^{-26}+2568q^{-27}+587q^{-28}-1580q^{-29}-2404q^{-30}-1523q^{-31}+240q^{-32}+1622q^{-33}+1758q^{-34}+727q^{-35}-636q^{-36}-1384q^{-37}-1133q^{-38}-177q^{-39}+727q^{-40}+1013q^{-41}+611q^{-42}-112q^{-43}-634q^{-44}-641q^{-45}-233q^{-46}+212q^{-47}+433q^{-48}+337q^{-49}+42q^{-50}-203q^{-51}-241q^{-52}-122q^{-53}+24q^{-54}+122q^{-55}+112q^{-56}+26q^{-57}-40q^{-58}-48q^{-59}-31q^{-60}-6q^{-61}+24q^{-62}+17q^{-63}+q^{-64}-3q^{-65}-3q^{-66}-5q^{-67}+q^{-68}+3q^{-69}-q^{-70}$
6	$q^{111}-3q^{110}+2q^{109}+q^{108}-2q^{107}+5q^{106}-6q^{105}+3q^{104}-9q^{103}+12q^{102}+5q^{101}-22q^{100}+19q^{99}-9q^{98}+10q^{97}-12q^{96}+33q^{95}-14q^{94}-95q^{93}+41q^{92}+39q^{91}+87q^{90}+27q^{89}+30q^{88}-203q^{87}-366q^{86}+86q^{85}+356q^{84}+542q^{83}+269q^{82}-209q^{81}-1128q^{80}-1401q^{79}+122q^{78}+1687q^{77}+2547q^{76}+1501q^{75}-1118q^{74}-4487q^{73}-5207q^{72}-720q^{71}+5346q^{70}+9172q^{69}+6757q^{68}-1874q^{67}-12759q^{66}-16513q^{65}-6740q^{64}+10371q^{63}+23970q^{62}+22592q^{61}+3544q^{60}-24102q^{59}-39357q^{58}-26368q^{57}+8130q^{56}+42949q^{55}+52636q^{54}+25440q^{53}-26293q^{52}-66239q^{51}-61777q^{50}-13345q^{49}+49816q^{48}+84951q^{47}+63530q^{46}-6489q^{45}-77209q^{44}-96492q^{43}-50565q^{42}+32363q^{41}+97465q^{40}+97537q^{39}+27996q^{38}-62409q^{37}-108472q^{36}-80956q^{35}+1832q^{34}+84205q^{33}+107686q^{32}+54096q^{31}-36325q^{30}-96530q^{29}-88864q^{28}-20008q^{27}+61840q^{26}+97531q^{25}+61125q^{24}-17903q^{23}-78153q^{22}-81755q^{21}-27420q^{20}+45793q^{19}+84006q^{18}+58906q^{17}-8820q^{16}-64998q^{15}-74653q^{14}-30977q^{13}+34853q^{12}+75008q^{11}+59566q^{10}+948q^{9}-53786q^{8}-71787q^{7}-39619q^{6}+20066q^{5}+65590q^{4}+64028q^{3}+17438q^{2}-36353q-66601-51600q^{-1}-2564q^{-2}+47771q^{-3}+64244q^{-4}+36874q^{-5}-9827q^{-6}-50697q^{-7}-57425q^{-8}-27811q^{-9}+19166q^{-10}+51100q^{-11}+48281q^{-12}+19289q^{-13}-21990q^{-14}-47116q^{-15}-43104q^{-16}-12262q^{-17}+23013q^{-18}+40860q^{-19}+36660q^{-20}+9674q^{-21}-20087q^{-22}-37102q^{-23}-30257q^{-24}-7793q^{-25}+15632q^{-26}+30926q^{-27}+26259q^{-28}+8683q^{-29}-13154q^{-30}-24182q^{-31}-21983q^{-32}-9668q^{-33}+8435q^{-34}+18881q^{-35}+19433q^{-36}+8574q^{-37}-3753q^{-38}-13447q^{-39}-16202q^{-40}-9093q^{-41}+653q^{-42}+9795q^{-43}+11599q^{-44}+9093q^{-45}+1870q^{-46}-5920q^{-47}-8909q^{-48}-7892q^{-49}-2526q^{-50}+2047q^{-51}+6356q^{-52}+6486q^{-53}+3254q^{-54}-539q^{-55}-3828q^{-56}-4309q^{-57}-3701q^{-58}-444q^{-59}+2026q^{-60}+3043q^{-61}+2648q^{-62}+1015q^{-63}-477q^{-64}-2135q^{-65}-1818q^{-66}-1032q^{-67}+74q^{-68}+900q^{-69}+1172q^{-70}+997q^{-71}+22q^{-72}-356q^{-73}-642q^{-74}-520q^{-75}-256q^{-76}+110q^{-77}+389q^{-78}+225q^{-79}+166q^{-80}-15q^{-81}-102q^{-82}-160q^{-83}-86q^{-84}+24q^{-85}+17q^{-86}+54q^{-87}+32q^{-88}+18q^{-89}-22q^{-90}-20q^{-91}+q^{-92}-7q^{-93}+3q^{-94}+3q^{-95}+5q^{-96}-q^{-97}-3q^{-98}+q^{-99}$
7	Not Available

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, 108]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 4, 17, 3], X[20, 13, 1, 14], X[14, 7, 15, 8], X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[12, 6, 13, 5], X[4, 12, 5, 11], X[2, 16, 3, 15]]
In[3]:=	GaussCode[Knot[10, 108]]
Out[3]=	GaussCode[1, -10, 2, -9, 8, -1, 4, -5, 6, -7, 9, -8, 3, -4, 10, -2, 7, -6, 5, -3]
In[4]:=	DTCode[Knot[10, 108]]
Out[4]=	DTCode[6, 16, 12, 14, 18, 4, 20, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 108]]
Out[5]=	BR[4, {1, 1, -2, 1, 1, 3, -2, 1, -2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 108]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 108]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 108]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 108]][t]
Out[10]=	2 8 14 2 3 -15 + -- - -- + -- + 14 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 108]][z]
Out[11]=	4 6 1 + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 108], Knot[11, NonAlternating, 161]}
In[13]:=	{KnotDet[Knot[10, 108]], KnotSignature[Knot[10, 108]]}
Out[13]=	{63, 2}
In[14]:=	Jones[Knot[10, 108]][q]
Out[14]=	-4 3 5 8 2 3 4 5 6 -9 - q + -- - -- + - + 10 q - 10 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 108]}
In[16]:=	A2Invariant[Knot[10, 108]][q]
Out[16]=	-12 -10 2 -2 4 6 8 10 16 18 2 - q + q + -- - q - q + q - 2 q + 2 q + q - q 4 q
In[17]:=	HOMFLYPT[Knot[10, 108]][a, z]
Out[17]=	2 2 4 4 6 2 2 z 2 z 2 2 4 z 3 z 2 4 6 z 1 + 2 z - ---- + ---- - 2 a z + 3 z - -- + ---- - a z + z + -- 4 2 4 2 2 a a a a a
In[18]:=	Kauffman[Knot[10, 108]][a, z]
Out[18]=	2 2 3 2 z 6 z 3 2 z 2 z 2 2 z 1 - --- - --- - 6 a z - 2 a z - 10 z - -- + ---- - 7 a z + -- - 3 a 6 4 7 a a a a 3 3 3 4 4 3 z 10 z 28 z 3 3 3 4 3 z 9 z ---- + ----- + ----- + 19 a z + 5 a z + 33 z + ---- - ---- + 5 3 a 6 4 a a a a 4 5 5 5 4 z 2 4 5 z 17 z 29 z 5 3 5 6 ---- + 17 a z + ---- - ----- - ----- - 11 a z - 4 a z - 33 z + 2 5 3 a a a a 6 6 7 7 7 z 13 z 2 6 8 z 4 z 7 3 7 8 ---- - ----- - 13 a z + ---- + ---- - 3 a z + a z + 9 z + 4 2 3 a a a a 8 9 6 z 2 8 2 z 9 ---- + 3 a z + ---- + 2 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 108]], Vassiliev[3][Knot[10, 108]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 108]][q, t]
Out[20]=	3 1 2 1 3 2 5 3 6 q + 5 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 4 5 q 3 5 5 2 7 2 7 3 9 3 --- + --- + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t t 9 4 11 4 13 5 q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 108], 2][q]
Out[21]=	-13 3 -11 11 9 14 30 5 39 43 11 62 32 + q - --- - q + --- - -- - -- + -- - -- - -- + -- + -- - -- + 12 10 9 8 7 6 5 4 3 2 q q q q q q q q q q 42 2 3 4 5 6 7 8 -- - 73 q + 30 q + 47 q - 67 q + 13 q + 47 q - 47 q + 2 q + q 9 10 11 12 13 14 15 16 17 30 q - 25 q + 3 q + 11 q - 11 q + 4 q + 2 q - 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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