10 138

10_137

10_139

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10 138 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X7,15,8,14 X15,7,16,6 X20,18,1,17 X18,13,19,14 X12,19,13,20
Gauss code	1, -4, 3, -1, 2, 7, -6, -3, 4, -2, 5, -10, 9, 6, -7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code	4 8 10 -14 2 16 18 -6 20 12
Conway Notation	[211,211,2-]

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][-7] Hyperbolic Volume 10.4672 A-Polynomial See Data:10 138/A-polynomial

[edit Notes for 10 138'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 10 138's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+8t-7+8t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, 2 }
Jones polynomial	$2q^{5}-4q^{4}+5q^{3}-6q^{2}+6q-5+4q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+5z^{2}a^{-2}-3z^{2}a^{-4}-6z^{2}+2a^{2}+3a^{-2}-2a^{-4}+a^{-6}-3$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+2az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+a^{2}z^{6}+3z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}-7az^{5}-14z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}-4a^{2}z^{4}-13z^{4}a^{-2}-5z^{4}a^{-4}-12z^{4}+6az^{3}+8z^{3}a^{-1}+5z^{3}a^{-3}+3z^{3}a^{-5}+5a^{2}z^{2}+10z^{2}a^{-2}+6z^{2}a^{-4}+3z^{2}a^{-6}+12z^{2}-az-za^{-1}-2za^{-3}-2za^{-5}-2a^{2}-3a^{-2}-2a^{-4}-a^{-6}-3$
The A2 invariant	$q^{10}+q^{8}+q^{4}-q^{2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
The G2 invariant	$q^{46}-q^{44}+4q^{42}-5q^{40}+5q^{38}-2q^{36}-4q^{34}+14q^{32}-18q^{30}+20q^{28}-11q^{26}-4q^{24}+19q^{22}-27q^{20}+29q^{18}-16q^{16}+17q^{12}-25q^{10}+19q^{8}-5q^{6}-13q^{4}+20q^{2}-21+7q^{-2}+8q^{-4}-25q^{-6}+33q^{-8}-29q^{-10}+14q^{-12}+5q^{-14}-25q^{-16}+35q^{-18}-34q^{-20}+24q^{-22}-4q^{-24}-12q^{-26}+28q^{-28}-27q^{-30}+17q^{-32}+q^{-34}-15q^{-36}+22q^{-38}-17q^{-40}+16q^{-44}-24q^{-46}+25q^{-48}-15q^{-50}-5q^{-52}+18q^{-54}-26q^{-56}+22q^{-58}-13q^{-60}+q^{-62}+9q^{-64}-12q^{-66}+11q^{-68}-8q^{-70}+5q^{-72}+q^{-74}-3q^{-76}+q^{-78}-2q^{-80}+2q^{-82}-q^{-84}+2q^{-86}+q^{-88}$

Further Quantum Invariants

Further quantum knot invariants for 10_138.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+2q^{3}-q+q^{-1}-q^{-5}+q^{-7}-2q^{-9}+2q^{-11}$
2	$q^{22}-q^{20}-2q^{18}+4q^{16}+q^{14}-6q^{12}+4q^{10}+5q^{8}-7q^{6}-q^{4}+7q^{2}-3-4q^{-2}+5q^{-4}+q^{-6}-5q^{-8}+q^{-10}+6q^{-12}-2q^{-14}-5q^{-16}+7q^{-18}+q^{-20}-8q^{-22}+4q^{-24}+2q^{-26}-4q^{-28}+q^{-30}+q^{-32}$
3	$q^{45}-q^{43}-2q^{41}+5q^{37}+3q^{35}-7q^{33}-8q^{31}+6q^{29}+14q^{27}-19q^{23}-8q^{21}+17q^{19}+19q^{17}-10q^{15}-26q^{13}+q^{11}+28q^{9}+10q^{7}-27q^{5}-19q^{3}+22q+24q^{-1}-15q^{-3}-25q^{-5}+11q^{-7}+28q^{-9}-6q^{-11}-24q^{-13}-q^{-15}+22q^{-17}+7q^{-19}-18q^{-21}-17q^{-23}+10q^{-25}+24q^{-27}+q^{-29}-28q^{-31}-13q^{-33}+28q^{-35}+21q^{-37}-21q^{-39}-24q^{-41}+14q^{-43}+21q^{-45}-3q^{-47}-17q^{-49}+8q^{-53}-2q^{-57}-2q^{-59}+2q^{-61}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{8}+q^{4}-q^{2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
2,0	$q^{28}+q^{26}-2q^{22}+3q^{18}+q^{16}-3q^{14}-q^{12}+3q^{10}+2q^{8}-3q^{6}+3q^{2}-2q^{-2}-q^{-6}-2q^{-8}+q^{-10}+2q^{-16}+7q^{-18}+2q^{-20}-4q^{-22}+q^{-26}-3q^{-28}-4q^{-30}+2q^{-34}+2q^{-36}-q^{-48}+q^{-52}$

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+4q^{16}-4q^{14}+6q^{12}-6q^{10}+7q^{8}-6q^{6}+4q^{4}-2q^{2}-3+5q^{-2}-8q^{-4}+11q^{-6}-12q^{-8}+14q^{-10}-11q^{-12}+9q^{-14}-6q^{-16}+2q^{-18}-4q^{-22}+5q^{-24}-7q^{-26}+7q^{-28}-6q^{-30}+5q^{-32}-3q^{-34}+3q^{-36}$
1,0	$q^{34}-q^{30}-q^{28}+3q^{26}+3q^{24}-2q^{22}-4q^{20}+q^{18}+6q^{16}+3q^{14}-6q^{12}-5q^{10}+4q^{8}+7q^{6}-q^{4}-8q^{2}-2+5q^{-2}+3q^{-4}-3q^{-6}-4q^{-8}+2q^{-10}+4q^{-12}-q^{-14}-5q^{-16}+q^{-18}+6q^{-20}+q^{-22}-5q^{-24}-3q^{-26}+5q^{-28}+5q^{-30}-3q^{-32}-6q^{-34}+2q^{-36}+7q^{-38}+q^{-40}-6q^{-42}-4q^{-44}+3q^{-46}+5q^{-48}-q^{-50}-4q^{-52}-2q^{-54}+q^{-56}+3q^{-58}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+4q^{42}-5q^{40}+5q^{38}-2q^{36}-4q^{34}+14q^{32}-18q^{30}+20q^{28}-11q^{26}-4q^{24}+19q^{22}-27q^{20}+29q^{18}-16q^{16}+17q^{12}-25q^{10}+19q^{8}-5q^{6}-13q^{4}+20q^{2}-21+7q^{-2}+8q^{-4}-25q^{-6}+33q^{-8}-29q^{-10}+14q^{-12}+5q^{-14}-25q^{-16}+35q^{-18}-34q^{-20}+24q^{-22}-4q^{-24}-12q^{-26}+28q^{-28}-27q^{-30}+17q^{-32}+q^{-34}-15q^{-36}+22q^{-38}-17q^{-40}+16q^{-44}-24q^{-46}+25q^{-48}-15q^{-50}-5q^{-52}+18q^{-54}-26q^{-56}+22q^{-58}-13q^{-60}+q^{-62}+9q^{-64}-12q^{-66}+11q^{-68}-8q^{-70}+5q^{-72}+q^{-74}-3q^{-76}+q^{-78}-2q^{-80}+2q^{-82}-q^{-84}+2q^{-86}+q^{-88}$

.

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["10 138"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+8t-7+8t^{-1}-5t^{-2}+t^{-3}$

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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}+2az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+a^{2}z^{6}+3z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}-7az^{5}-14z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}-4a^{2}z^{4}-13z^{4}a^{-2}-5z^{4}a^{-4}-12z^{4}+6az^{3}+8z^{3}a^{-1}+5z^{3}a^{-3}+3z^{3}a^{-5}+5a^{2}z^{2}+10z^{2}a^{-2}+6z^{2}a^{-4}+3z^{2}a^{-6}+12z^{2}-az-za^{-1}-2za^{-3}-2za^{-5}-2a^{2}-3a^{-2}-2a^{-4}-a^{-6}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n117, ...}

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 $-12$ $-16$ $72$ $66$ $30$ $192$ ${\frac {704}{3}}$ ${\frac {128}{3}}$ $48$ $-288$ $128$ $-792$ $-360$ $-{\frac {3311}{10}}$ ${\frac {382}{5}}$ $-{\frac {6142}{15}}$ ${\frac {431}{6}}$ $-{\frac {1071}{10}}$

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 10 138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{15}-5q^{13}+7q^{12}+2q^{11}-17q^{10}+16q^{9}+8q^{8}-29q^{7}+19q^{6}+16q^{5}-34q^{4}+13q^{3}+22q^{2}-30q+4+23q^{-1}-20q^{-2}-4q^{-3}+17q^{-4}-8q^{-5}-5q^{-6}+7q^{-7}-q^{-8}-2q^{-9}+q^{-10}$
3	$2q^{29}-4q^{28}+2q^{26}+10q^{25}-12q^{24}-17q^{23}+16q^{22}+34q^{21}-19q^{20}-55q^{19}+19q^{18}+76q^{17}-12q^{16}-96q^{15}+4q^{14}+105q^{13}+11q^{12}-110q^{11}-23q^{10}+104q^{9}+36q^{8}-95q^{7}-46q^{6}+81q^{5}+54q^{4}-61q^{3}-63q^{2}+45q+64-22q^{-1}-65q^{-2}+4q^{-3}+56q^{-4}+15q^{-5}-47q^{-6}-23q^{-7}+29q^{-8}+31q^{-9}-18q^{-10}-25q^{-11}+4q^{-12}+20q^{-13}+q^{-14}-11q^{-15}-4q^{-16}+6q^{-17}+2q^{-18}-q^{-19}-2q^{-20}+q^{-21}$
4	$q^{48}-5q^{46}+11q^{44}+3q^{43}-6q^{42}-29q^{41}-7q^{40}+56q^{39}+34q^{38}-17q^{37}-113q^{36}-58q^{35}+145q^{34}+140q^{33}+4q^{32}-253q^{31}-201q^{30}+215q^{29}+311q^{28}+107q^{27}-367q^{26}-400q^{25}+200q^{24}+440q^{23}+259q^{22}-376q^{21}-547q^{20}+120q^{19}+456q^{18}+371q^{17}-302q^{16}-582q^{15}+38q^{14}+376q^{13}+414q^{12}-193q^{11}-538q^{10}-31q^{9}+257q^{8}+413q^{7}-72q^{6}-448q^{5}-96q^{4}+114q^{3}+378q^{2}+57q-315-140q^{-1}-40q^{-2}+285q^{-3}+153q^{-4}-145q^{-5}-117q^{-6}-156q^{-7}+136q^{-8}+159q^{-9}+2q^{-10}-25q^{-11}-170q^{-12}+78q^{-14}+56q^{-15}+59q^{-16}-96q^{-17}-46q^{-18}-3q^{-19}+26q^{-20}+68q^{-21}-21q^{-22}-22q^{-23}-23q^{-24}-6q^{-25}+32q^{-26}+2q^{-27}-9q^{-29}-8q^{-30}+7q^{-31}+q^{-32}+2q^{-33}-q^{-34}-2q^{-35}+q^{-36}$
5	$2q^{71}-4q^{70}+2q^{67}+12q^{66}+2q^{65}-28q^{64}-20q^{63}+8q^{62}+38q^{61}+73q^{60}+6q^{59}-123q^{58}-140q^{57}-16q^{56}+187q^{55}+299q^{54}+100q^{53}-316q^{52}-530q^{51}-235q^{50}+404q^{49}+833q^{48}+502q^{47}-451q^{46}-1189q^{45}-861q^{44}+414q^{43}+1507q^{42}+1294q^{41}-244q^{40}-1769q^{39}-1743q^{38}-q^{37}+1904q^{36}+2127q^{35}+318q^{34}-1915q^{33}-2422q^{32}-619q^{31}+1817q^{30}+2580q^{29}+893q^{28}-1660q^{27}-2628q^{26}-1082q^{25}+1461q^{24}+2577q^{23}+1219q^{22}-1262q^{21}-2476q^{20}-1292q^{19}+1064q^{18}+2326q^{17}+1350q^{16}-856q^{15}-2166q^{14}-1398q^{13}+639q^{12}+1983q^{11}+1437q^{10}-384q^{9}-1769q^{8}-1486q^{7}+117q^{6}+1520q^{5}+1487q^{4}+173q^{3}-1204q^{2}-1465q-441+865q^{-1}+1334q^{-2}+675q^{-3}-484q^{-4}-1144q^{-5}-811q^{-6}+128q^{-7}+853q^{-8}+845q^{-9}+180q^{-10}-542q^{-11}-745q^{-12}-375q^{-13}+209q^{-14}+569q^{-15}+458q^{-16}+29q^{-17}-328q^{-18}-402q^{-19}-207q^{-20}+113q^{-21}+295q^{-22}+237q^{-23}+48q^{-24}-133q^{-25}-213q^{-26}-127q^{-27}+23q^{-28}+123q^{-29}+133q^{-30}+53q^{-31}-49q^{-32}-93q^{-33}-71q^{-34}-9q^{-35}+51q^{-36}+60q^{-37}+22q^{-38}-11q^{-39}-33q^{-40}-30q^{-41}-2q^{-42}+19q^{-43}+13q^{-44}+6q^{-45}-11q^{-47}-6q^{-48}+3q^{-49}+2q^{-50}+q^{-51}+2q^{-52}-q^{-53}-2q^{-54}+q^{-55}$
6	$q^{99}-5q^{97}+7q^{95}+4q^{94}-q^{92}-10q^{91}-33q^{90}-10q^{89}+59q^{88}+67q^{87}+20q^{86}-31q^{85}-122q^{84}-195q^{83}-62q^{82}+257q^{81}+407q^{80}+252q^{79}-79q^{78}-560q^{77}-895q^{76}-468q^{75}+628q^{74}+1434q^{73}+1313q^{72}+304q^{71}-1360q^{70}-2689q^{69}-2062q^{68}+516q^{67}+3106q^{66}+3836q^{65}+2149q^{64}-1635q^{63}-5303q^{62}-5388q^{61}-1267q^{60}+4189q^{59}+7192q^{58}+5816q^{57}-106q^{56}-7123q^{55}-9375q^{54}-4840q^{53}+3324q^{52}+9467q^{51}+9864q^{50}+3090q^{49}-6812q^{48}-11955q^{47}-8532q^{46}+850q^{45}+9496q^{44}+12299q^{43}+6222q^{42}-4911q^{41}-12255q^{40}-10591q^{39}-1587q^{38}+8003q^{37}+12603q^{36}+7896q^{35}-2919q^{34}-11135q^{33}-10849q^{32}-2970q^{31}+6298q^{30}+11702q^{29}+8245q^{28}-1516q^{27}-9694q^{26}-10261q^{25}-3638q^{24}+4811q^{23}+10513q^{22}+8180q^{21}-286q^{20}-8186q^{19}-9566q^{18}-4371q^{17}+3130q^{16}+9158q^{15}+8220q^{14}+1363q^{13}-6217q^{12}-8719q^{11}-5424q^{10}+830q^{9}+7204q^{8}+8087q^{7}+3427q^{6}-3442q^{5}-7140q^{4}-6256q^{3}-1911q^{2}+4286q+7006+5123q^{-1}-154q^{-2}-4390q^{-3}-5880q^{-4}-4121q^{-5}+780q^{-6}+4481q^{-7}+5297q^{-8}+2493q^{-9}-950q^{-10}-3787q^{-11}-4539q^{-12}-1980q^{-13}+1169q^{-14}+3515q^{-15}+3167q^{-16}+1655q^{-17}-857q^{-18}-2896q^{-19}-2648q^{-20}-1220q^{-21}+916q^{-22}+1813q^{-23}+2148q^{-24}+1081q^{-25}-601q^{-26}-1432q^{-27}-1556q^{-28}-641q^{-29}+27q^{-30}+1024q^{-31}+1156q^{-32}+582q^{-33}-26q^{-34}-604q^{-35}-600q^{-36}-665q^{-37}-45q^{-38}+337q^{-39}+442q^{-40}+379q^{-41}+118q^{-42}-21q^{-43}-382q^{-44}-249q^{-45}-127q^{-46}+25q^{-47}+135q^{-48}+168q^{-49}+188q^{-50}-49q^{-51}-62q^{-52}-105q^{-53}-77q^{-54}-40q^{-55}+26q^{-56}+102q^{-57}+21q^{-58}+24q^{-59}-13q^{-60}-24q^{-61}-39q^{-62}-16q^{-63}+24q^{-64}+3q^{-65}+14q^{-66}+5q^{-67}+3q^{-68}-11q^{-69}-8q^{-70}+5q^{-71}-2q^{-72}+2q^{-73}+q^{-74}+2q^{-75}-q^{-76}-2q^{-77}+q^{-78}$
7	$2q^{131}-4q^{130}+10q^{126}+4q^{125}-4q^{124}-16q^{123}-22q^{122}-4q^{121}+12q^{120}+34q^{119}+78q^{118}+53q^{117}-55q^{116}-161q^{115}-201q^{114}-82q^{113}+110q^{112}+332q^{111}+519q^{110}+367q^{109}-196q^{108}-861q^{107}-1194q^{106}-828q^{105}+220q^{104}+1498q^{103}+2463q^{102}+2111q^{101}+138q^{100}-2584q^{99}-4616q^{98}-4296q^{97}-1200q^{96}+3502q^{95}+7564q^{94}+8110q^{93}+3737q^{92}-3940q^{91}-11278q^{90}-13524q^{89}-8126q^{88}+3039q^{87}+14879q^{86}+20312q^{85}+14871q^{84}+11q^{83}-17596q^{82}-27866q^{81}-23558q^{80}-5459q^{79}+18330q^{78}+34791q^{77}+33434q^{76}+13450q^{75}-16471q^{74}-40128q^{73}-43282q^{72}-22929q^{71}+11993q^{70}+42776q^{69}+51651q^{68}+32875q^{67}-5362q^{66}-42614q^{65}-57642q^{64}-41859q^{63}-2210q^{62}+39922q^{61}+60703q^{60}+48891q^{59}+9628q^{58}-35597q^{57}-61109q^{56}-53422q^{55}-15874q^{54}+30643q^{53}+59505q^{52}+55554q^{51}+20410q^{50}-25991q^{49}-56715q^{48}-55711q^{47}-23237q^{46}+22058q^{45}+53549q^{44}+54696q^{43}+24672q^{42}-19075q^{41}-50470q^{40}-53065q^{39}-25281q^{38}+16697q^{37}+47676q^{36}+51393q^{35}+25669q^{34}-14603q^{33}-45126q^{32}-49896q^{31}-26220q^{30}+12340q^{29}+42479q^{28}+48603q^{27}+27295q^{26}-9483q^{25}-39537q^{24}-47428q^{23}-28846q^{22}+5877q^{21}+35879q^{20}+45981q^{19}+30851q^{18}-1327q^{17}-31309q^{16}-44046q^{15}-32974q^{14}-3902q^{13}+25631q^{12}+41104q^{11}+34720q^{10}+9706q^{9}-18807q^{8}-36959q^{7}-35645q^{6}-15403q^{5}+11148q^{4}+31224q^{3}+35068q^{2}+20477q-2969-24086q^{-1}-32665q^{-2}-24040q^{-3}-4890q^{-4}+15793q^{-5}+28117q^{-6}+25535q^{-7}+11601q^{-8}-7124q^{-9}-21699q^{-10}-24446q^{-11}-16263q^{-12}-1023q^{-13}+14079q^{-14}+20880q^{-15}+18220q^{-16}+7466q^{-17}-6217q^{-18}-15283q^{-19}-17356q^{-20}-11524q^{-21}-640q^{-22}+8867q^{-23}+14043q^{-24}+12633q^{-25}+5489q^{-26}-2628q^{-27}-9241q^{-28}-11298q^{-29}-7856q^{-30}-2056q^{-31}+4272q^{-32}+8104q^{-33}+7715q^{-34}+4766q^{-35}-93q^{-36}-4413q^{-37}-5942q^{-38}-5309q^{-39}-2389q^{-40}+1154q^{-41}+3337q^{-42}+4322q^{-43}+3255q^{-44}+957q^{-45}-1012q^{-46}-2621q^{-47}-2798q^{-48}-1757q^{-49}-543q^{-50}+966q^{-51}+1729q^{-52}+1595q^{-53}+1179q^{-54}+141q^{-55}-678q^{-56}-979q^{-57}-1080q^{-58}-580q^{-59}-48q^{-60}+299q^{-61}+691q^{-62}+603q^{-63}+327q^{-64}+83q^{-65}-282q^{-66}-339q^{-67}-312q^{-68}-270q^{-69}+2q^{-70}+146q^{-71}+203q^{-72}+222q^{-73}+72q^{-74}+9q^{-75}-48q^{-76}-149q^{-77}-99q^{-78}-53q^{-79}+6q^{-80}+73q^{-81}+40q^{-82}+40q^{-83}+37q^{-84}-15q^{-85}-27q^{-86}-34q^{-87}-22q^{-88}+13q^{-89}+q^{-90}+5q^{-91}+16q^{-92}+5q^{-93}+2q^{-94}-8q^{-95}-8q^{-96}+3q^{-97}-2q^{-99}+2q^{-100}+q^{-101}+2q^{-102}-q^{-103}-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, 138]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 12, 17, 11], X[7, 15, 8, 14], X[15, 7, 16, 6], X[20, 18, 1, 17], X[18, 13, 19, 14], X[12, 19, 13, 20]]
In[3]:=	GaussCode[Knot[10, 138]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, 7, -6, -3, 4, -2, 5, -10, 9, 6, -7, -5, 8, -9, 10, -8]
In[4]:=	DTCode[Knot[10, 138]]
Out[4]=	DTCode[4, 8, 10, -14, 2, 16, 18, -6, 20, 12]
In[5]:=	br = BR[Knot[10, 138]]
Out[5]=	BR[5, {-1, 2, -1, 2, 3, 2, 2, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[10, 138]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 138]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 138]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 138]][t]
Out[10]=	-3 5 8 2 3 -7 + t - -- + - + 8 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[10, 138]][z]
Out[11]=	2 4 6 1 - 3 z + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 138]}
In[13]:=	{KnotDet[Knot[10, 138]], KnotSignature[Knot[10, 138]]}
Out[13]=	{35, 2}
In[14]:=	Jones[Knot[10, 138]][q]
Out[14]=	-3 2 4 2 3 4 5 -5 + q - -- + - + 6 q - 6 q + 5 q - 4 q + 2 q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 138], Knot[11, NonAlternating, 117]}
In[16]:=	A2Invariant[Knot[10, 138]][q]
Out[16]=	-10 -8 -4 -2 4 6 8 10 12 14 16 20 q + q + q - q - q + 2 q - q + q - q - q + q + q
In[17]:=	HOMFLYPT[Knot[10, 138]][a, z]
Out[17]=	2 2 4 -6 2 3 2 2 3 z 5 z 2 2 4 z -3 + a - -- + -- + 2 a - 6 z - ---- + ---- + a z - 2 z - -- + 4 2 4 2 4 a a a a a 4 6 4 z z ---- + -- 2 2 a a
In[18]:=	Kauffman[Knot[10, 138]][a, z]
Out[18]=	2 2 -6 2 3 2 2 z 2 z z 2 3 z 6 z -3 - a - -- - -- - 2 a - --- - --- - - - a z + 12 z + ---- + ---- + 4 2 5 3 a 6 4 a a a a a a 2 3 3 3 4 10 z 2 2 3 z 5 z 8 z 3 4 5 z ----- + 5 a z + ---- + ---- + ---- + 6 a z - 12 z - ---- - 2 5 3 a 4 a a a a 4 5 5 5 6 6 13 z 2 4 z 6 z 14 z 5 6 3 z 3 z ----- - 4 a z + -- - ---- - ----- - 7 a z + z + ---- + ---- + 2 5 3 a 4 2 a a a a a 7 7 8 2 6 3 z 5 z 7 8 z a z + ---- + ---- + 2 a z + z + -- 3 a 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 138]], Vassiliev[3][Knot[10, 138]]}
Out[19]=	{-3, -2}
In[20]:=	Kh[Knot[10, 138]][q, t]
Out[20]=	3 1 1 1 3 1 2 3 q 4 q + 3 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 11 4 3 q t + 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + 2 q t
In[21]:=	ColouredJones[Knot[10, 138], 2][q]
Out[21]=	-10 2 -8 7 5 8 17 4 20 23 2 4 + q - -- - q + -- - -- - -- + -- - -- - -- + -- - 30 q + 22 q + 9 7 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 10 13 q - 34 q + 16 q + 19 q - 29 q + 8 q + 16 q - 17 q + 11 12 13 15 2 q + 7 q - 5 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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