K11a131

K11a130

K11a132

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a131 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a131/ThurstonBennequinNumber Hyperbolic Volume 16.9146 A-Polynomial See Data:K11a131/A-polynomial

[edit Notes for K11a131's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 2

[edit Notes for K11a131's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+27t-31+27t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 131, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-14+19q^{-1}-21q^{-2}+21q^{-3}-18q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+11a^{4}z^{2}-10a^{2}z^{2}+3z^{2}-2a^{6}+5a^{4}-4a^{2}+2$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+12a^{3}z^{9}+6az^{9}+7a^{6}z^{8}+12a^{4}z^{8}+12a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-7a^{5}z^{7}-23a^{3}z^{7}-7az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-12a^{6}z^{6}-40a^{4}z^{6}-42a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-6a^{7}z^{5}+3a^{5}z^{5}+11a^{3}z^{5}-8az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+13a^{6}z^{4}+50a^{4}z^{4}+44a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-2a^{9}z^{3}+a^{7}z^{3}+3a^{5}z^{3}+3a^{3}z^{3}+8az^{3}+5z^{3}a^{-1}+2a^{8}z^{2}-8a^{6}z^{2}-26a^{4}z^{2}-22a^{2}z^{2}+z^{2}a^{-2}-5z^{2}+a^{9}z-2a^{5}z-2a^{3}z-2az-za^{-1}+2a^{6}+5a^{4}+4a^{2}+2$
The A2 invariant	$-q^{24}-2q^{18}+4q^{16}-2q^{14}+2q^{12}+2q^{10}-3q^{8}+4q^{6}-5q^{4}+3q^{2}-q^{-2}+3q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-8q^{118}+q^{116}+12q^{114}-27q^{112}+44q^{110}-56q^{108}+54q^{106}-36q^{104}-6q^{102}+67q^{100}-134q^{98}+189q^{96}-215q^{94}+175q^{92}-65q^{90}-116q^{88}+328q^{86}-483q^{84}+513q^{82}-369q^{80}+57q^{78}+313q^{76}-618q^{74}+723q^{72}-554q^{70}+171q^{68}+276q^{66}-586q^{64}+623q^{62}-357q^{60}-81q^{58}+486q^{56}-662q^{54}+512q^{52}-88q^{50}-432q^{48}+841q^{46}-941q^{44}+692q^{42}-174q^{40}-429q^{38}+894q^{36}-1061q^{34}+873q^{32}-404q^{30}-173q^{28}+650q^{26}-849q^{24}+707q^{22}-295q^{20}-203q^{18}+556q^{16}-628q^{14}+376q^{12}+72q^{10}-503q^{8}+729q^{6}-637q^{4}+269q^{2}+211-607q^{-2}+766q^{-4}-646q^{-6}+328q^{-8}+51q^{-10}-350q^{-12}+482q^{-14}-435q^{-16}+280q^{-18}-84q^{-20}-71q^{-22}+150q^{-24}-163q^{-26}+123q^{-28}-66q^{-30}+20q^{-32}+12q^{-34}-24q^{-36}+22q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a131 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["K11a131"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+27t-31+27t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 131, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-14+19q^{-1}-21q^{-2}+21q^{-3}-18q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+11a^{4}z^{2}-10a^{2}z^{2}+3z^{2}-2a^{6}+5a^{4}-4a^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+12a^{3}z^{9}+6az^{9}+7a^{6}z^{8}+12a^{4}z^{8}+12a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-7a^{5}z^{7}-23a^{3}z^{7}-7az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-12a^{6}z^{6}-40a^{4}z^{6}-42a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-6a^{7}z^{5}+3a^{5}z^{5}+11a^{3}z^{5}-8az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+13a^{6}z^{4}+50a^{4}z^{4}+44a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-2a^{9}z^{3}+a^{7}z^{3}+3a^{5}z^{3}+3a^{3}z^{3}+8az^{3}+5z^{3}a^{-1}+2a^{8}z^{2}-8a^{6}z^{2}-26a^{4}z^{2}-22a^{2}z^{2}+z^{2}a^{-2}-5z^{2}+a^{9}z-2a^{5}z-2a^{3}z-2az-za^{-1}+2a^{6}+5a^{4}+4a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a252, K11a254,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a218,}

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["K11a131"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+27t-31+27t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+9q-14+19q^{-1}-21q^{-2}+21q^{-3}-18q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{K11a252, K11a254,}

In[6]:=

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

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

Out[6]=

{K11a218,}

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-24$ $8$ ${\frac {206}{3}}$ ${\frac {10}{3}}$ $-96$ $-176$ $0$ $8$ ${\frac {32}{3}}$ $288$ ${\frac {824}{3}}$ ${\frac {40}{3}}$ ${\frac {24031}{30}}$ ${\frac {4138}{15}}$ $-{\frac {4858}{45}}$ $-{\frac {127}{18}}$ $-{\frac {1889}{30}}$

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 $s=$-2 is the signature of K11a131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a130 K11a132