K11n143

K11n142

K11n144

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n143 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X5,16,6,17 X22,8,1,7 X9,18,10,19 X2,12,3,11 X13,21,14,20 X15,10,16,11 X17,6,18,7 X19,15,20,14 X8,22,9,21
Gauss code	1, -6, 2, -1, -3, 9, 4, -11, -5, 8, 6, -2, -7, 10, -8, 3, -9, 5, -10, 7, 11, -4
Dowker-Thistlethwaite code	4 12 -16 22 -18 2 -20 -10 -6 -14 8

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n143/ThurstonBennequinNumber Hyperbolic Volume 9.31377 A-Polynomial See Data:K11n143/A-polynomial

[edit Notes for K11n143's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n143's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+2t^{2}-t+1-t^{-1}+2t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-4z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$q^{6}-2q^{5}+2q^{4}-2q^{3}+2q^{2}-q+1-q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-5z^{4}a^{-2}+z^{4}a^{-4}+a^{2}z^{2}-6z^{2}a^{-2}+3z^{2}a^{-4}+a^{2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+az^{7}+z^{7}a^{-1}+2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-6z^{6}a^{-2}-4z^{6}a^{-4}+z^{6}a^{-6}-6az^{5}-7z^{5}a^{-1}-10z^{5}a^{-3}-9z^{5}a^{-5}-5a^{2}z^{4}+10z^{4}a^{-2}+3z^{4}a^{-4}-4z^{4}a^{-6}-2z^{4}+8az^{3}+11z^{3}a^{-1}+12z^{3}a^{-3}+9z^{3}a^{-5}+5a^{2}z^{2}-7z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}-2az-4za^{-1}-4za^{-3}-2za^{-5}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	Data:K11n143/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n143/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n143 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["K11n143"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+2t^{2}-t+1-t^{-1}+2t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}a^{-4}+az^{7}+z^{7}a^{-1}+2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-6z^{6}a^{-2}-4z^{6}a^{-4}+z^{6}a^{-6}-6az^{5}-7z^{5}a^{-1}-10z^{5}a^{-3}-9z^{5}a^{-5}-5a^{2}z^{4}+10z^{4}a^{-2}+3z^{4}a^{-4}-4z^{4}a^{-6}-2z^{4}+8az^{3}+11z^{3}a^{-1}+12z^{3}a^{-3}+9z^{3}a^{-5}+5a^{2}z^{2}-7z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}-2az-4za^{-1}-4za^{-3}-2za^{-5}-a^{2}+a^{-2}+a^{-4}$

"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["K11n143"];

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]=

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

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:

(-2, -1)

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 $-8$ $-8$ $32$ ${\frac {212}{3}}$ ${\frac {148}{3}}$ $64$ ${\frac {496}{3}}$ ${\frac {160}{3}}$ $56$ $-{\frac {256}{3}}$ $32$ $-{\frac {1696}{3}}$ $-{\frac {1184}{3}}$ $-{\frac {5431}{15}}$ ${\frac {5524}{15}}$ $-{\frac {37924}{45}}$ ${\frac {1543}{9}}$ $-{\frac {3511}{15}}$

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 K11n143. 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                   1   -1 9                 1 1   0 7             1 2 1     0 5             1 1       0 3         1 2 2         1 1       1 2 1           0 -1       1 2             1 -3   1 1 1               -1 -5                       0 -7 1                     1

Integral Khovanov Homology (db, data source)

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

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.

K11n142 K11n144