K11a154

K11a153

K11a155

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a154 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a154/ThurstonBennequinNumber Hyperbolic Volume 11.5462 A-Polynomial See Data:K11a154/A-polynomial

[edit Notes for K11a154's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a154's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t^{2}+17t-25+17t^{-1}-4t^{-2}$
Conway polynomial	$-4z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 67, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+7q^{5}-9q^{4}+11q^{3}-10q^{2}+9q-7+4q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-2z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}+a^{2}+a^{-4}+a^{-6}-a^{-8}-1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+4z^{9}a^{-3}+2z^{9}a^{-5}-z^{8}a^{-2}-z^{8}a^{-4}+2z^{8}a^{-6}+2z^{8}+2az^{7}-5z^{7}a^{-1}-14z^{7}a^{-3}-5z^{7}a^{-5}+2z^{7}a^{-7}+a^{2}z^{6}-z^{6}a^{-2}-2z^{6}a^{-6}+2z^{6}a^{-8}-4z^{6}-7az^{5}+3z^{5}a^{-1}+22z^{5}a^{-3}+9z^{5}a^{-5}-2z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+2z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}-5z^{4}a^{-8}-3z^{4}+6az^{3}-11z^{3}a^{-3}-5z^{3}a^{-5}-3z^{3}a^{-7}-3z^{3}a^{-9}+4a^{2}z^{2}-z^{2}a^{-2}-3z^{2}a^{-4}+2z^{2}a^{-6}+3z^{2}a^{-8}+5z^{2}-2az-2za^{-1}+za^{-3}+za^{-5}+2za^{-7}+2za^{-9}-a^{2}+a^{-4}-a^{-6}-a^{-8}-1$
The A2 invariant	Data:K11a154/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a154/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a154 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["K11a154"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t^{2}+17t-25+17t^{-1}-4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+2q^{7}-4q^{6}+7q^{5}-9q^{4}+11q^{3}-10q^{2}+9q-7+4q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+4z^{9}a^{-3}+2z^{9}a^{-5}-z^{8}a^{-2}-z^{8}a^{-4}+2z^{8}a^{-6}+2z^{8}+2az^{7}-5z^{7}a^{-1}-14z^{7}a^{-3}-5z^{7}a^{-5}+2z^{7}a^{-7}+a^{2}z^{6}-z^{6}a^{-2}-2z^{6}a^{-6}+2z^{6}a^{-8}-4z^{6}-7az^{5}+3z^{5}a^{-1}+22z^{5}a^{-3}+9z^{5}a^{-5}-2z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+2z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}-5z^{4}a^{-8}-3z^{4}+6az^{3}-11z^{3}a^{-3}-5z^{3}a^{-5}-3z^{3}a^{-7}-3z^{3}a^{-9}+4a^{2}z^{2}-z^{2}a^{-2}-3z^{2}a^{-4}+2z^{2}a^{-6}+3z^{2}a^{-8}+5z^{2}-2az-2za^{-1}+za^{-3}+za^{-5}+2za^{-7}+2za^{-9}-a^{2}+a^{-4}-a^{-6}-a^{-8}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_30,}

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["K11a154"];

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]=

{ $-4t^{2}+17t-25+17t^{-1}-4t^{-2}$, $-q^{8}+2q^{7}-4q^{6}+7q^{5}-9q^{4}+11q^{3}-10q^{2}+9q-7+4q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{10_30,}

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:

(1, 5)

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 $4$ $40$ $8$ ${\frac {446}{3}}$ ${\frac {106}{3}}$ $160$ ${\frac {1744}{3}}$ ${\frac {160}{3}}$ $168$ ${\frac {32}{3}}$ $800$ ${\frac {1784}{3}}$ ${\frac {424}{3}}$ ${\frac {87631}{30}}$ $-{\frac {1954}{5}}$ ${\frac {74582}{45}}$ ${\frac {2129}{18}}$ ${\frac {6991}{30}}$

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 K11a154. 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                     1   1 13                   3 1   -2 11                 4 1     3 9               5 3       -2 7             6 4         2 5           4 5           1 3         5 6             -1 1       3 5               2 -1     1 4                 -3 -3   1 3                   2 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a153 K11a155