10 119

10_118

10_120

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10 119 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X7,18,8,19 X3948 X17,3,18,2 X5,15,6,14 X9,17,10,16 X15,11,16,10 X11,5,12,4 X13,20,14,1 X19,12,20,13
Gauss code	-1, 4, -3, 8, -5, 1, -2, 3, -6, 7, -8, 10, -9, 5, -7, 6, -4, 2, -10, 9
Dowker-Thistlethwaite code	6 8 14 18 16 4 20 10 2 12
Conway Notation	[8*2:.20]

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-7] Hyperbolic Volume 15.9387 A-Polynomial See Data:10 119/A-polynomial

[edit Notes for 10 119'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 10 119's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 101, 0 }
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-12q^{3}+16q^{2}-17q+16-13q^{-1}+9q^{-2}-4q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$3z^{9}a^{-1}+3z^{9}a^{-3}+15z^{8}a^{-2}+6z^{8}a^{-4}+9z^{8}+12az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+9a^{2}z^{6}-31z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+4a^{3}z^{5}-17az^{5}-37z^{5}a^{-1}-26z^{5}a^{-3}-10z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+13z^{4}a^{-2}+8z^{4}a^{-4}-2z^{4}a^{-6}-7z^{4}-a^{3}z^{3}+9az^{3}+22z^{3}a^{-1}+19z^{3}a^{-3}+7z^{3}a^{-5}+4a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-6}+6z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1$
The A2 invariant	$q^{12}-2q^{10}+2q^{8}+2q^{6}-3q^{4}+3q^{2}-3+q^{-2}+q^{-4}-q^{-6}+4q^{-8}-3q^{-10}+q^{-12}+q^{-14}-2q^{-16}+q^{-18}$
The G2 invariant	$q^{66}-3q^{64}+6q^{62}-10q^{60}+11q^{58}-10q^{56}+4q^{54}+14q^{52}-35q^{50}+60q^{48}-78q^{46}+71q^{44}-44q^{42}-17q^{40}+105q^{38}-184q^{36}+238q^{34}-223q^{32}+127q^{30}+33q^{28}-222q^{26}+371q^{24}-410q^{22}+305q^{20}-82q^{18}-176q^{16}+370q^{14}-400q^{12}+264q^{10}-15q^{8}-242q^{6}+363q^{4}-299q^{2}+59+250q^{-2}-475q^{-4}+516q^{-6}-332q^{-8}-q^{-10}+350q^{-12}-598q^{-14}+642q^{-16}-473q^{-18}+155q^{-20}+204q^{-22}-467q^{-24}+561q^{-26}-441q^{-28}+178q^{-30}+115q^{-32}-337q^{-34}+382q^{-36}-246q^{-38}-7q^{-40}+274q^{-42}-413q^{-44}+359q^{-46}-130q^{-48}-174q^{-50}+415q^{-52}-496q^{-54}+387q^{-56}-150q^{-58}-116q^{-60}+311q^{-62}-370q^{-64}+302q^{-66}-149q^{-68}-7q^{-70}+108q^{-72}-148q^{-74}+125q^{-76}-73q^{-78}+27q^{-80}+8q^{-82}-22q^{-84}+21q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 10_119.

A1 Invariants.

Weight	Invariant
1	$q^{9}-3q^{7}+5q^{5}-4q^{3}+3q-q^{-1}-q^{-3}+4q^{-5}-4q^{-7}+4q^{-9}-3q^{-11}+q^{-13}$
2	$q^{26}-3q^{24}+2q^{22}+8q^{20}-18q^{18}+4q^{16}+31q^{14}-38q^{12}-9q^{10}+54q^{8}-32q^{6}-29q^{4}+47q^{2}-33q^{-2}+10q^{-4}+28q^{-6}-19q^{-8}-29q^{-10}+41q^{-12}+8q^{-14}-52q^{-16}+31q^{-18}+31q^{-20}-48q^{-22}+4q^{-24}+32q^{-26}-20q^{-28}-9q^{-30}+13q^{-32}-q^{-34}-3q^{-36}+q^{-38}$
3	$q^{51}-3q^{49}+2q^{47}+5q^{45}-6q^{43}-11q^{41}+12q^{39}+32q^{37}-32q^{35}-68q^{33}+55q^{31}+132q^{29}-55q^{27}-237q^{25}+26q^{23}+343q^{21}+57q^{19}-417q^{17}-186q^{15}+425q^{13}+319q^{11}-350q^{9}-411q^{7}+215q^{5}+441q^{3}-53q-408q^{-1}-100q^{-3}+332q^{-5}+225q^{-7}-236q^{-9}-317q^{-11}+136q^{-13}+386q^{-15}-30q^{-17}-435q^{-19}-77q^{-21}+448q^{-23}+195q^{-25}-420q^{-27}-316q^{-29}+346q^{-31}+404q^{-33}-214q^{-35}-444q^{-37}+63q^{-39}+411q^{-41}+73q^{-43}-313q^{-45}-158q^{-47}+185q^{-49}+175q^{-51}-74q^{-53}-135q^{-55}+2q^{-57}+78q^{-59}+24q^{-61}-34q^{-63}-17q^{-65}+8q^{-67}+8q^{-69}-q^{-71}-3q^{-73}+q^{-75}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}-2q^{10}+2q^{8}+2q^{6}-3q^{4}+3q^{2}-3+q^{-2}+q^{-4}-q^{-6}+4q^{-8}-3q^{-10}+q^{-12}+q^{-14}-2q^{-16}+q^{-18}$
1,1	$q^{36}-6q^{34}+18q^{32}-38q^{30}+77q^{28}-150q^{26}+258q^{24}-404q^{22}+617q^{20}-892q^{18}+1198q^{16}-1522q^{14}+1824q^{12}-2026q^{10}+2036q^{8}-1802q^{6}+1301q^{4}-514q^{2}-480+1584q^{-2}-2643q^{-4}+3530q^{-6}-4132q^{-8}+4358q^{-10}-4205q^{-12}+3680q^{-14}-2842q^{-16}+1796q^{-18}-674q^{-20}-386q^{-22}+1298q^{-24}-1946q^{-26}+2274q^{-28}-2298q^{-30}+2086q^{-32}-1722q^{-34}+1284q^{-36}-874q^{-38}+540q^{-40}-296q^{-42}+145q^{-44}-62q^{-46}+22q^{-48}-6q^{-50}+q^{-52}$
2,0	$q^{32}-2q^{30}+6q^{26}-3q^{24}-10q^{22}+5q^{20}+17q^{18}-6q^{16}-23q^{14}+8q^{12}+24q^{10}-15q^{8}-19q^{6}+19q^{4}+13q^{2}-10-6q^{-2}+13q^{-4}-4q^{-6}-10q^{-8}+11q^{-10}-2q^{-12}-16q^{-14}+9q^{-16}+20q^{-18}-14q^{-20}-14q^{-22}+16q^{-24}+15q^{-26}-15q^{-28}-16q^{-30}+15q^{-32}+9q^{-34}-10q^{-36}-7q^{-38}+4q^{-40}+6q^{-42}-2q^{-44}-2q^{-46}+q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-3q^{26}+10q^{22}-11q^{20}-7q^{18}+27q^{16}-18q^{14}-17q^{12}+41q^{10}-19q^{8}-23q^{6}+38q^{4}-11q^{2}-18+16q^{-2}+5q^{-4}-6q^{-6}-13q^{-8}+16q^{-10}+12q^{-12}-33q^{-14}+15q^{-16}+25q^{-18}-38q^{-20}+12q^{-22}+23q^{-24}-30q^{-26}+10q^{-28}+11q^{-30}-14q^{-32}+5q^{-34}+2q^{-36}-3q^{-38}+q^{-40}$
1,0,0	$q^{15}-2q^{13}+3q^{11}-q^{9}+3q^{7}-3q^{5}+3q^{3}-3q+2q^{-7}-q^{-9}+4q^{-11}-3q^{-13}+2q^{-15}-2q^{-17}+2q^{-19}-2q^{-21}+q^{-23}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{34}-2q^{32}-2q^{30}+7q^{28}+q^{26}-11q^{24}+q^{22}+17q^{20}-2q^{18}-22q^{16}+10q^{14}+26q^{12}-20q^{10}-25q^{8}+30q^{6}+13q^{4}-31q^{2}+4+31q^{-2}-12q^{-4}-23q^{-6}+24q^{-8}+9q^{-10}-36q^{-12}+6q^{-14}+35q^{-16}-23q^{-18}-24q^{-20}+34q^{-22}+13q^{-24}-30q^{-26}-3q^{-28}+26q^{-30}-3q^{-32}-20q^{-34}+8q^{-36}+12q^{-38}-9q^{-40}-4q^{-42}+6q^{-44}-q^{-46}-2q^{-48}+q^{-50}$
1,0,0,0	$q^{18}-2q^{16}+3q^{14}+3q^{8}-3q^{6}+3q^{4}-3q^{2}-q^{-2}+q^{-8}+2q^{-10}-q^{-12}+4q^{-14}-3q^{-16}+2q^{-18}-q^{-20}-q^{-22}+2q^{-24}-2q^{-26}+q^{-28}$

B2 Invariants.

Weight	Invariant
0,1	$q^{28}-3q^{26}+6q^{24}-12q^{22}+21q^{20}-29q^{18}+39q^{16}-46q^{14}+49q^{12}-45q^{10}+35q^{8}-19q^{6}-2q^{4}+27q^{2}-52+72q^{-2}-87q^{-4}+94q^{-6}-91q^{-8}+80q^{-10}-60q^{-12}+37q^{-14}-11q^{-16}-11q^{-18}+30q^{-20}-42q^{-22}+49q^{-24}-48q^{-26}+42q^{-28}-35q^{-30}+24q^{-32}-15q^{-34}+8q^{-36}-3q^{-38}+q^{-40}$
1,0	$q^{46}-3q^{42}-3q^{40}+3q^{38}+11q^{36}+4q^{34}-16q^{32}-18q^{30}+8q^{28}+33q^{26}+11q^{24}-35q^{22}-33q^{20}+21q^{18}+49q^{16}+3q^{14}-49q^{12}-24q^{10}+37q^{8}+37q^{6}-22q^{4}-40q^{2}+8+39q^{-2}+4q^{-4}-35q^{-6}-11q^{-8}+30q^{-10}+17q^{-12}-26q^{-14}-23q^{-16}+24q^{-18}+32q^{-20}-17q^{-22}-44q^{-24}+3q^{-26}+50q^{-28}+20q^{-30}-43q^{-32}-41q^{-34}+24q^{-36}+50q^{-38}-42q^{-42}-20q^{-44}+25q^{-46}+26q^{-48}-8q^{-50}-19q^{-52}-3q^{-54}+10q^{-56}+5q^{-58}-3q^{-60}-3q^{-62}+q^{-66}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{38}-3q^{36}+3q^{34}-4q^{32}+11q^{30}-16q^{28}+16q^{26}-21q^{24}+33q^{22}-35q^{20}+33q^{18}-37q^{16}+41q^{14}-30q^{12}+22q^{10}-18q^{8}+5q^{6}+15q^{4}-25q^{2}+35-52q^{-2}+64q^{-4}-66q^{-6}+70q^{-8}-75q^{-10}+69q^{-12}-58q^{-14}+52q^{-16}-42q^{-18}+25q^{-20}-8q^{-22}-q^{-24}+12q^{-26}-25q^{-28}+35q^{-30}-36q^{-32}+38q^{-34}-39q^{-36}+36q^{-38}-30q^{-40}+25q^{-42}-20q^{-44}+13q^{-46}-8q^{-48}+5q^{-50}-3q^{-52}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-3q^{64}+6q^{62}-10q^{60}+11q^{58}-10q^{56}+4q^{54}+14q^{52}-35q^{50}+60q^{48}-78q^{46}+71q^{44}-44q^{42}-17q^{40}+105q^{38}-184q^{36}+238q^{34}-223q^{32}+127q^{30}+33q^{28}-222q^{26}+371q^{24}-410q^{22}+305q^{20}-82q^{18}-176q^{16}+370q^{14}-400q^{12}+264q^{10}-15q^{8}-242q^{6}+363q^{4}-299q^{2}+59+250q^{-2}-475q^{-4}+516q^{-6}-332q^{-8}-q^{-10}+350q^{-12}-598q^{-14}+642q^{-16}-473q^{-18}+155q^{-20}+204q^{-22}-467q^{-24}+561q^{-26}-441q^{-28}+178q^{-30}+115q^{-32}-337q^{-34}+382q^{-36}-246q^{-38}-7q^{-40}+274q^{-42}-413q^{-44}+359q^{-46}-130q^{-48}-174q^{-50}+415q^{-52}-496q^{-54}+387q^{-56}-150q^{-58}-116q^{-60}+311q^{-62}-370q^{-64}+302q^{-66}-149q^{-68}-7q^{-70}+108q^{-72}-148q^{-74}+125q^{-76}-73q^{-78}+27q^{-80}+8q^{-82}-22q^{-84}+21q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

.

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["10 119"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$q^{6}-4q^{5}+8q^{4}-12q^{3}+16q^{2}-17q+16-13q^{-1}+9q^{-2}-4q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{9}a^{-1}+3z^{9}a^{-3}+15z^{8}a^{-2}+6z^{8}a^{-4}+9z^{8}+12az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+9a^{2}z^{6}-31z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+4a^{3}z^{5}-17az^{5}-37z^{5}a^{-1}-26z^{5}a^{-3}-10z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+13z^{4}a^{-2}+8z^{4}a^{-4}-2z^{4}a^{-6}-7z^{4}-a^{3}z^{3}+9az^{3}+22z^{3}a^{-1}+19z^{3}a^{-3}+7z^{3}a^{-5}+4a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-6}+6z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a84, ...}

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

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 $-4$ $0$ $8$ ${\frac {82}{3}}$ ${\frac {62}{3}}$ $0$ $32$ $0$ $32$ $-{\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {248}{3}}$ $-{\frac {1951}{30}}$ ${\frac {1022}{15}}$ $-{\frac {9902}{45}}$ ${\frac {1855}{18}}$ $-{\frac {1471}{30}}$

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 10 119. 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 13                     1 1 11                   3   -3 9                 5 1   4 7               7 3     -4 5             9 5       4 3           8 7         -1 1         8 9           -1 -1       6 9             3 -3     3 7               -4 -5   1 6                 5 -7   3                   -3 -9 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{18}-4q^{17}+2q^{16}+15q^{15}-26q^{14}-9q^{13}+67q^{12}-54q^{11}-61q^{10}+146q^{9}-54q^{8}-144q^{7}+206q^{6}-21q^{5}-214q^{4}+216q^{3}+26q^{2}-232q+173+59q^{-1}-185q^{-2}+97q^{-3}+56q^{-4}-99q^{-5}+34q^{-6}+27q^{-7}-30q^{-8}+7q^{-9}+5q^{-10}-4q^{-11}+q^{-12}$
3	$q^{36}-4q^{35}+2q^{34}+9q^{33}+q^{32}-29q^{31}-15q^{30}+67q^{29}+55q^{28}-105q^{27}-152q^{26}+128q^{25}+304q^{24}-95q^{23}-495q^{22}-27q^{21}+690q^{20}+243q^{19}-843q^{18}-534q^{17}+920q^{16}+861q^{15}-901q^{14}-1196q^{13}+816q^{12}+1476q^{11}-648q^{10}-1721q^{9}+458q^{8}+1881q^{7}-232q^{6}-1971q^{5}+5q^{4}+1962q^{3}+229q^{2}-1864q-427+1654q^{-1}+584q^{-2}-1370q^{-3}-653q^{-4}+1028q^{-5}+645q^{-6}-701q^{-7}-547q^{-8}+417q^{-9}+414q^{-10}-227q^{-11}-261q^{-12}+100q^{-13}+151q^{-14}-45q^{-15}-74q^{-16}+23q^{-17}+28q^{-18}-9q^{-19}-10q^{-20}+3q^{-21}+5q^{-22}-4q^{-23}+q^{-24}$
4	$q^{60}-4q^{59}+2q^{58}+9q^{57}-5q^{56}-2q^{55}-35q^{54}+9q^{53}+78q^{52}+24q^{51}-2q^{50}-234q^{49}-123q^{48}+256q^{47}+344q^{46}+343q^{45}-653q^{44}-912q^{43}-60q^{42}+923q^{41}+1908q^{40}-284q^{39}-2244q^{38}-2063q^{37}+229q^{36}+4410q^{35}+2383q^{34}-2044q^{33}-5254q^{32}-3503q^{31}+5323q^{30}+6610q^{29}+1610q^{28}-6835q^{27}-9280q^{26}+2669q^{25}+9461q^{24}+7637q^{23}-4976q^{22}-14134q^{21}-2500q^{20}+9273q^{19}+13279q^{18}-726q^{17}-16431q^{16}-7818q^{15}+6885q^{14}+17045q^{13}+3954q^{12}-16512q^{11}-12057q^{10}+3581q^{9}+18858q^{8}+8212q^{7}-14791q^{6}-14952q^{5}-359q^{4}+18417q^{3}+11734q^{2}-10923q-15659-4667q^{-1}+14854q^{-2}+13309q^{-3}-5243q^{-4}-12982q^{-5}-7653q^{-6}+8713q^{-7}+11442q^{-8}-153q^{-9}-7596q^{-10}-7376q^{-11}+2925q^{-12}+6895q^{-13}+1895q^{-14}-2602q^{-15}-4508q^{-16}+81q^{-17}+2698q^{-18}+1345q^{-19}-242q^{-20}-1752q^{-21}-309q^{-22}+666q^{-23}+403q^{-24}+151q^{-25}-454q^{-26}-84q^{-27}+124q^{-28}+38q^{-29}+63q^{-30}-91q^{-31}-2q^{-32}+27q^{-33}-8q^{-34}+11q^{-35}-14q^{-36}+3q^{-37}+5q^{-38}-4q^{-39}+q^{-40}$
5	$q^{90}-4q^{89}+2q^{88}+9q^{87}-5q^{86}-8q^{85}-8q^{84}-11q^{83}+20q^{82}+71q^{81}+29q^{80}-74q^{79}-149q^{78}-157q^{77}+43q^{76}+386q^{75}+535q^{74}+142q^{73}-629q^{72}-1233q^{71}-964q^{70}+508q^{69}+2323q^{68}+2741q^{67}+615q^{66}-3068q^{65}-5523q^{64}-3818q^{63}+2301q^{62}+8706q^{61}+9376q^{60}+1607q^{59}-10271q^{58}-16735q^{57}-10010q^{56}+7863q^{55}+23717q^{54}+22560q^{53}+908q^{52}-26863q^{51}-37290q^{50}-16878q^{49}+22875q^{48}+50391q^{47}+38553q^{46}-9576q^{45}-57548q^{44}-62501q^{43}-12860q^{42}+55504q^{41}+84185q^{40}+41841q^{39}-43024q^{38}-99477q^{37}-73282q^{36}+21200q^{35}+105906q^{34}+103113q^{33}+6837q^{32}-103332q^{31}-127723q^{30}-37512q^{29}+92962q^{28}+146068q^{27}+67383q^{26}-77858q^{25}-157705q^{24}-94208q^{23}+60130q^{22}+164281q^{21}+117290q^{20}-42344q^{19}-166996q^{18}-136672q^{17}+24919q^{16}+167338q^{15}+153331q^{14}-8048q^{13}-165525q^{12}-167906q^{11}-9396q^{10}+161160q^{9}+180507q^{8}+28179q^{7}-152563q^{6}-190371q^{5}-48880q^{4}+138440q^{3}+195494q^{2}+70496q-117398-193539q^{-1}-91167q^{-2}+90096q^{-3}+182525q^{-4}+107326q^{-5}-58453q^{-6}-161747q^{-7}-115969q^{-8}+26501q^{-9}+132782q^{-10}+114703q^{-11}+1431q^{-12}-99257q^{-13}-103670q^{-14}-21353q^{-15}+65902q^{-16}+85046q^{-17}+31853q^{-18}-37406q^{-19}-63145q^{-20}-33162q^{-21}+16513q^{-22}+41826q^{-23}+28465q^{-24}-3794q^{-25}-24745q^{-26}-20758q^{-27}-2018q^{-28}+12681q^{-29}+13217q^{-30}+3546q^{-31}-5625q^{-32}-7396q^{-33}-2904q^{-34}+2103q^{-35}+3599q^{-36}+1790q^{-37}-611q^{-38}-1566q^{-39}-928q^{-40}+176q^{-41}+615q^{-42}+361q^{-43}-45q^{-44}-198q^{-45}-133q^{-46}-q^{-47}+96q^{-48}+34q^{-49}-33q^{-50}-9q^{-51}+2q^{-52}-9q^{-53}+12q^{-54}+7q^{-55}-14q^{-56}+3q^{-57}+5q^{-58}-4q^{-59}+q^{-60}$
6	$q^{126}-4q^{125}+2q^{124}+9q^{123}-5q^{122}-8q^{121}-14q^{120}+16q^{119}+13q^{117}+76q^{116}-19q^{115}-87q^{114}-173q^{113}-37q^{112}+32q^{111}+212q^{110}+597q^{109}+324q^{108}-193q^{107}-1077q^{106}-1170q^{105}-1074q^{104}+156q^{103}+2687q^{102}+3644q^{101}+2879q^{100}-1066q^{99}-4766q^{98}-8642q^{97}-7628q^{96}+740q^{95}+10641q^{94}+18227q^{93}+14755q^{92}+3147q^{91}-19152q^{90}-35574q^{89}-31743q^{88}-7345q^{87}+31458q^{86}+59060q^{85}+63137q^{84}+18940q^{83}-48250q^{82}-100827q^{81}-104108q^{80}-39317q^{79}+63176q^{78}+162419q^{77}+167622q^{76}+70659q^{75}-92693q^{74}-232947q^{73}-253500q^{72}-121546q^{71}+132952q^{70}+331550q^{69}+362644q^{68}+165312q^{67}-169952q^{66}-455762q^{65}-502014q^{64}-207811q^{63}+237928q^{62}+607679q^{61}+632205q^{60}+258986q^{59}-338753q^{58}-797916q^{57}-763179q^{56}-261150q^{55}+483253q^{54}+980938q^{53}+900348q^{52}+205544q^{51}-692790q^{50}-1177502q^{49}-962377q^{48}-71166q^{47}+920165q^{46}+1389200q^{45}+936713q^{44}-174355q^{43}-1196663q^{42}-1510284q^{41}-793359q^{40}+480296q^{39}+1513405q^{38}+1523101q^{37}+489665q^{36}-882073q^{35}-1734843q^{34}-1385111q^{33}-80578q^{32}+1351700q^{31}+1832666q^{30}+1041271q^{29}-468324q^{28}-1721154q^{27}-1748009q^{26}-549293q^{25}+1105672q^{24}+1949534q^{23}+1414470q^{22}-119797q^{21}-1634951q^{20}-1964344q^{19}-894103q^{18}+889871q^{17}+2005998q^{16}+1695002q^{15}+170021q^{14}-1543644q^{13}-2140097q^{12}-1212056q^{11}+657080q^{10}+2026626q^{9}+1970649q^{8}+515939q^{7}-1359164q^{6}-2260711q^{5}-1577469q^{4}+273280q^{3}+1883026q^{2}+2187926q+976970-931323q^{-1}-2157307q^{-2}-1893519q^{-3}-296676q^{-4}+1414673q^{-5}+2136498q^{-6}+1404434q^{-7}-260131q^{-8}-1666401q^{-9}-1908538q^{-10}-843657q^{-11}+667373q^{-12}+1658528q^{-13}+1507905q^{-14}+382318q^{-15}-886523q^{-16}-1479924q^{-17}-1046864q^{-18}-30361q^{-19}+905971q^{-20}+1169119q^{-21}+662338q^{-22}-186409q^{-23}-807055q^{-24}-821489q^{-25}-355208q^{-26}+263383q^{-27}+622264q^{-28}+537974q^{-29}+145088q^{-30}-263289q^{-31}-421230q^{-32}-312460q^{-33}-34549q^{-34}+205459q^{-35}+267735q^{-36}+157234q^{-37}-20098q^{-38}-133588q^{-39}-149712q^{-40}-70701q^{-41}+29488q^{-42}+83394q^{-43}+72557q^{-44}+22346q^{-45}-21110q^{-46}-43985q^{-47}-32343q^{-48}-4874q^{-49}+15681q^{-50}+19492q^{-51}+10333q^{-52}+734q^{-53}-8165q^{-54}-8255q^{-55}-2932q^{-56}+1772q^{-57}+3263q^{-58}+2009q^{-59}+1067q^{-60}-1037q^{-61}-1411q^{-62}-502q^{-63}+221q^{-64}+369q^{-65}+109q^{-66}+266q^{-67}-120q^{-68}-199q^{-69}-16q^{-70}+55q^{-71}+42q^{-72}-52q^{-73}+49q^{-74}-5q^{-75}-34q^{-76}+11q^{-77}+8q^{-78}+7q^{-79}-14q^{-80}+3q^{-81}+5q^{-82}-4q^{-83}+q^{-84}$
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, 119]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 18, 8, 19], X[3, 9, 4, 8], X[17, 3, 18, 2], X[5, 15, 6, 14], X[9, 17, 10, 16], X[15, 11, 16, 10], X[11, 5, 12, 4], X[13, 20, 14, 1], X[19, 12, 20, 13]]
In[3]:=	GaussCode[Knot[10, 119]]
Out[3]=	GaussCode[-1, 4, -3, 8, -5, 1, -2, 3, -6, 7, -8, 10, -9, 5, -7, 6, -4, 2, -10, 9]
In[4]:=	DTCode[Knot[10, 119]]
Out[4]=	DTCode[6, 8, 14, 18, 16, 4, 20, 10, 2, 12]
In[5]:=	br = BR[Knot[10, 119]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, -1, 2, 3, 3, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 119]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 119]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 119]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 119]][t]
Out[10]=	2 10 23 2 3 31 - -- + -- - -- - 23 t + 10 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 119]][z]
Out[11]=	2 4 6 1 - z - 2 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 119], Knot[11, Alternating, 84]}
In[13]:=	{KnotDet[Knot[10, 119]], KnotSignature[Knot[10, 119]]}
Out[13]=	{101, 0}
In[14]:=	Jones[Knot[10, 119]][q]
Out[14]=	-4 4 9 13 2 3 4 5 6 16 + q - -- + -- - -- - 17 q + 16 q - 12 q + 8 q - 4 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 119]}
In[16]:=	A2Invariant[Knot[10, 119]][q]
Out[16]=	-12 2 2 2 3 3 2 4 6 8 10 -3 + q - --- + -- + -- - -- + -- + q + q - q + 4 q - 3 q + 10 8 6 4 2 q q q q q 12 14 16 18 q + q - 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 119]][a, z]
Out[17]=	2 2 4 4 -2 2 2 z z 2 2 4 z 2 z 2 4 -1 + a + a - 2 z + -- - -- + a z - 2 z + -- - ---- + a z - 4 2 4 2 a a a a 6 6 z z - -- 2 a
In[18]:=	Kauffman[Knot[10, 119]][a, z]
Out[18]=	2 2 -2 2 z 3 z 4 z 2 z z 2 2 -1 - a - a - -- - --- - --- - 2 a z + 6 z + -- + -- + 4 a z + 5 3 a 6 2 a a a a 3 3 3 4 4 4 7 z 19 z 22 z 3 3 3 4 2 z 8 z 13 z ---- + ----- + ----- + 9 a z - a z - 7 z - ---- + ---- + ----- - 5 3 a 6 4 2 a a a a a 5 5 5 2 4 4 4 10 z 26 z 37 z 5 3 5 6 9 a z + a z - ----- - ----- - ----- - 17 a z + 4 a z - 7 z + 5 3 a a a 6 6 6 7 7 7 z 14 z 31 z 2 6 4 z 5 z 13 z 7 8 -- - ----- - ----- + 9 a z + ---- + ---- + ----- + 12 a z + 9 z + 6 4 2 5 3 a a a a a a 8 8 9 9 6 z 15 z 3 z 3 z ---- + ----- + ---- + ---- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 119]], Vassiliev[3][Knot[10, 119]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[10, 119]][q, t]
Out[20]=	9 1 3 1 6 3 7 6 - + 8 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 8 q t + 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + 5 q t + 9 5 11 5 13 6 q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 119], 2][q]
Out[21]=	-12 4 5 7 30 27 34 99 56 97 185 59 173 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 232 q + 26 q + 216 q - 214 q - 21 q + 206 q - 144 q - 54 q + 9 10 11 12 13 14 15 16 146 q - 61 q - 54 q + 67 q - 9 q - 26 q + 15 q + 2 q - 17 18 4 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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