K11a129

K11a128

K11a130

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a129 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X14,6,15,5 X20,7,21,8 X16,9,17,10 X2,11,3,12 X18,13,19,14 X22,16,1,15 X12,17,13,18 X6,19,7,20 X8,21,9,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -3, 8, -5, 9, -7, 10, -4, 11, -8
Dowker-Thistlethwaite code	4 10 14 20 16 2 18 22 12 6 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a129/ThurstonBennequinNumber Hyperbolic Volume 15.5089 A-Polynomial See Data:K11a129/A-polynomial

[edit Notes for K11a129's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 4

[edit Notes for K11a129's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+15t^{2}-22t+25-22t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 113, -4 }
Jones polynomial	$-q+4-7q^{-1}+12q^{-2}-15q^{-3}+18q^{-4}-18q^{-5}+15q^{-6}-12q^{-7}+7q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-10z^{2}a^{6}-5a^{6}+z^{8}a^{4}+5z^{6}a^{4}+9z^{4}a^{4}+8z^{2}a^{4}+3a^{4}-z^{6}a^{2}-3z^{4}a^{2}-z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-2z^{3}a^{11}+6z^{6}a^{10}-6z^{4}a^{10}+3z^{2}a^{10}+9z^{7}a^{9}-15z^{5}a^{9}+14z^{3}a^{9}-5za^{9}+9z^{8}a^{8}-15z^{6}a^{8}+10z^{4}a^{8}-4z^{2}a^{8}+2a^{8}+6z^{9}a^{7}-4z^{7}a^{7}-15z^{5}a^{7}+20z^{3}a^{7}-9za^{7}+2z^{10}a^{6}+10z^{8}a^{6}-41z^{6}a^{6}+43z^{4}a^{6}-22z^{2}a^{6}+5a^{6}+11z^{9}a^{5}-30z^{7}a^{5}+18z^{5}a^{5}+z^{3}a^{5}-3za^{5}+2z^{10}a^{4}+5z^{8}a^{4}-35z^{6}a^{4}+42z^{4}a^{4}-18z^{2}a^{4}+3a^{4}+5z^{9}a^{3}-16z^{7}a^{3}+12z^{5}a^{3}-z^{3}a^{3}+za^{3}+4z^{8}a^{2}-15z^{6}a^{2}+16z^{4}a^{2}-4z^{2}a^{2}-a^{2}+z^{7}a-3z^{5}a+2z^{3}a$
The A2 invariant	$q^{30}+2q^{24}-3q^{22}+q^{20}-3q^{18}-2q^{16}+2q^{14}-3q^{12}+5q^{10}-q^{8}+2q^{6}+2q^{4}-q^{2}+2-q^{-2}$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+6q^{154}-5q^{152}+9q^{148}-19q^{146}+29q^{144}-37q^{142}+33q^{140}-20q^{138}-6q^{136}+46q^{134}-80q^{132}+105q^{130}-107q^{128}+75q^{126}-22q^{124}-56q^{122}+137q^{120}-193q^{118}+212q^{116}-171q^{114}+77q^{112}+61q^{110}-193q^{108}+289q^{106}-295q^{104}+195q^{102}-23q^{100}-162q^{98}+283q^{96}-276q^{94}+149q^{92}+57q^{90}-242q^{88}+303q^{86}-219q^{84}-6q^{82}+257q^{80}-431q^{78}+431q^{76}-257q^{74}-33q^{72}+325q^{70}-516q^{68}+523q^{66}-362q^{64}+81q^{62}+206q^{60}-404q^{58}+453q^{56}-328q^{54}+106q^{52}+135q^{50}-300q^{48}+319q^{46}-185q^{44}-38q^{42}+255q^{40}-358q^{38}+300q^{36}-96q^{34}-158q^{32}+364q^{30}-427q^{28}+334q^{26}-126q^{24}-111q^{22}+284q^{20}-335q^{18}+273q^{16}-133q^{14}-9q^{12}+107q^{10}-144q^{8}+125q^{6}-74q^{4}+26q^{2}+11-26q^{-2}+23q^{-4}-17q^{-6}+8q^{-8}-3q^{-10}+q^{-12}$

Further Quantum Invariants

K11a129 Quantum Invariants.

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["K11a129"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+15t^{2}-22t+25-22t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}-z^{4}+1$

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]=

{ 113, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q+4-7q^{-1}+12q^{-2}-15q^{-3}+18q^{-4}-18q^{-5}+15q^{-6}-12q^{-7}+7q^{-8}-3q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-10z^{2}a^{6}-5a^{6}+z^{8}a^{4}+5z^{6}a^{4}+9z^{4}a^{4}+8z^{2}a^{4}+3a^{4}-z^{6}a^{2}-3z^{4}a^{2}-z^{2}a^{2}+a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-2z^{3}a^{11}+6z^{6}a^{10}-6z^{4}a^{10}+3z^{2}a^{10}+9z^{7}a^{9}-15z^{5}a^{9}+14z^{3}a^{9}-5za^{9}+9z^{8}a^{8}-15z^{6}a^{8}+10z^{4}a^{8}-4z^{2}a^{8}+2a^{8}+6z^{9}a^{7}-4z^{7}a^{7}-15z^{5}a^{7}+20z^{3}a^{7}-9za^{7}+2z^{10}a^{6}+10z^{8}a^{6}-41z^{6}a^{6}+43z^{4}a^{6}-22z^{2}a^{6}+5a^{6}+11z^{9}a^{5}-30z^{7}a^{5}+18z^{5}a^{5}+z^{3}a^{5}-3za^{5}+2z^{10}a^{4}+5z^{8}a^{4}-35z^{6}a^{4}+42z^{4}a^{4}-18z^{2}a^{4}+3a^{4}+5z^{9}a^{3}-16z^{7}a^{3}+12z^{5}a^{3}-z^{3}a^{3}+za^{3}+4z^{8}a^{2}-15z^{6}a^{2}+16z^{4}a^{2}-4z^{2}a^{2}-a^{2}+z^{7}a-3z^{5}a+2z^{3}a$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11a129"];

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]=

{ $t^{4}-6t^{3}+15t^{2}-22t+25-22t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$, $-q+4-7q^{-1}+12q^{-2}-15q^{-3}+18q^{-4}-18q^{-5}+15q^{-6}-12q^{-7}+7q^{-8}-3q^{-9}+q^{-10}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(0, 3)

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 $0$ $24$ $0$ $-96$ $8$ $0$ $144$ $-64$ $-40$ $0$ $288$ $0$ $0$ $608$ $-{\frac {136}{3}}$ ${\frac {2296}{3}}$ ${\frac {64}{3}}$ $80$

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 $s=$-4 is the signature of K11a129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     3   3 -1                   4 1   -3 -3                 8 3     5 -5               8 5       -3 -7             10 7         3 -9           8 8           0 -11         7 10             -3 -13       5 8               3 -15     2 7                 -5 -17   1 5                   4 -19   2                     -2 -21 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a128 K11a130