K11n152

K11n151

K11n153

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n152 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8493 X12,5,13,6 X2837 X9,19,10,18 X11,20,12,21 X4,13,5,14 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,17,22,16
Gauss code	1, -4, 2, -7, 3, -1, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, 6, -11, 9
Dowker-Thistlethwaite code	6 8 12 2 -18 -20 4 -10 -22 -14 -16

A Braid Representative
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:K11n152/ThurstonBennequinNumber Hyperbolic Volume 12.4339 A-Polynomial See Data:K11n152/A-polynomial

[edit Notes for K11n152's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,3]$ Rasmussen s-Invariant -2

[edit Notes for K11n152's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+6t-7+6t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, 2 }
Jones polynomial	$-q^{8}+3q^{7}-4q^{6}+5q^{5}-5q^{4}+4q^{3}-3q^{2}+q+1-q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}+z^{6}a^{-4}-7z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}+z^{4}-13z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}-7a^{-2}+5a^{-4}-a^{-6}+4$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+2z^{8}a^{-4}+z^{8}a^{-6}+z^{8}-7z^{7}a^{-1}-7z^{7}a^{-3}+3z^{7}a^{-5}+3z^{7}a^{-7}-17z^{6}a^{-2}-13z^{6}a^{-4}+3z^{6}a^{-8}-7z^{6}+13z^{5}a^{-1}+11z^{5}a^{-3}-11z^{5}a^{-5}-8z^{5}a^{-7}+z^{5}a^{-9}+40z^{4}a^{-2}+28z^{4}a^{-4}-5z^{4}a^{-6}-8z^{4}a^{-8}+15z^{4}-7z^{3}a^{-1}+2z^{3}a^{-3}+15z^{3}a^{-5}+4z^{3}a^{-7}-2z^{3}a^{-9}-31z^{2}a^{-2}-19z^{2}a^{-4}+2z^{2}a^{-6}+3z^{2}a^{-8}-13z^{2}-4za^{-3}-6za^{-5}-2za^{-7}+7a^{-2}+5a^{-4}+a^{-6}+4$
The A2 invariant	$q^{6}+q^{4}+2q^{2}+q^{-2}-2q^{-4}-2q^{-6}-2q^{-10}+2q^{-12}+2q^{-16}+q^{-18}-q^{-20}+q^{-22}-q^{-24}$
The G2 invariant	Data:K11n152/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n152 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["K11n152"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+6t-7+6t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 23, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-4q^{6}+5q^{5}-5q^{4}+4q^{3}-3q^{2}+q+1-q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}+z^{6}a^{-4}-7z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}+z^{4}-13z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}-7a^{-2}+5a^{-4}-a^{-6}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+2z^{8}a^{-4}+z^{8}a^{-6}+z^{8}-7z^{7}a^{-1}-7z^{7}a^{-3}+3z^{7}a^{-5}+3z^{7}a^{-7}-17z^{6}a^{-2}-13z^{6}a^{-4}+3z^{6}a^{-8}-7z^{6}+13z^{5}a^{-1}+11z^{5}a^{-3}-11z^{5}a^{-5}-8z^{5}a^{-7}+z^{5}a^{-9}+40z^{4}a^{-2}+28z^{4}a^{-4}-5z^{4}a^{-6}-8z^{4}a^{-8}+15z^{4}-7z^{3}a^{-1}+2z^{3}a^{-3}+15z^{3}a^{-5}+4z^{3}a^{-7}-2z^{3}a^{-9}-31z^{2}a^{-2}-19z^{2}a^{-4}+2z^{2}a^{-6}+3z^{2}a^{-8}-13z^{2}-4za^{-3}-6za^{-5}-2za^{-7}+7a^{-2}+5a^{-4}+a^{-6}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_6, K11n20, K11n151,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n151,}

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["K11n152"];

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]=

{ $-2t^{2}+6t-7+6t^{-1}-2t^{-2}$, $-q^{8}+3q^{7}-4q^{6}+5q^{5}-5q^{4}+4q^{3}-3q^{2}+q+1-q^{-1}+q^{-2}$ }

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]=

{8_6, K11n20, K11n151,}

In[6]:=

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

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

Out[6]=

{K11n151,}

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 $-8$ $-8$ $32$ ${\frac {212}{3}}$ ${\frac {100}{3}}$ $64$ ${\frac {400}{3}}$ ${\frac {64}{3}}$ $24$ $-{\frac {256}{3}}$ $32$ $-{\frac {1696}{3}}$ $-{\frac {800}{3}}$ $-{\frac {8791}{15}}$ ${\frac {1604}{15}}$ $-{\frac {29044}{45}}$ ${\frac {871}{9}}$ $-{\frac {2071}{15}}$

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=$2 is the signature of K11n152. 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 7 χ 17                       1 -1 15                     2   2 13                   2 1   -1 11                 3 2     1 9             1 3 2       0 7             2 3         -1 5         1 3 3           1 3       1 1 2             -2 1       1 3               2 -1   1 1                   0 -3                         0 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11n151 K11n153