K11a45

K11a44

K11a46

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a45 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,6,15,5 X2837 X20,9,21,10 X18,11,19,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X10,19,11,20 X16,22,17,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -6, 10, -5, 11, -8
Dowker-Thistlethwaite code	4 8 14 2 20 18 6 22 12 10 16

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

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a45/ThurstonBennequinNumber Hyperbolic Volume 13.2123 A-Polynomial See Data:K11a45/A-polynomial

[edit Notes for K11a45'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 K11a45's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-6t^{2}+22t-31+22t^{-1}-6t^{-2}$
Conway polynomial	$-6z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+10q^{5}-12q^{4}+14q^{3}-14q^{2}+11q-8+5q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-3z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}+a^{2}z^{2}-5z^{2}a^{-2}-z^{2}a^{-4}+3z^{2}a^{-6}+a^{2}-3a^{-2}+a^{-4}+2a^{-6}-a^{-8}+1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+4z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}+2az^{7}-9z^{7}a^{-1}-18z^{7}a^{-3}-3z^{7}a^{-5}+4z^{7}a^{-7}+a^{2}z^{6}-22z^{6}a^{-2}-19z^{6}a^{-4}-4z^{6}a^{-6}+3z^{6}a^{-8}-9z^{6}-6az^{5}+12z^{5}a^{-1}+25z^{5}a^{-3}+z^{5}a^{-5}-5z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+37z^{4}a^{-2}+28z^{4}a^{-4}-3z^{4}a^{-6}-6z^{4}a^{-8}+8z^{4}+3az^{3}-11z^{3}a^{-1}-14z^{3}a^{-3}-2z^{3}a^{-7}-2z^{3}a^{-9}+4a^{2}z^{2}-23z^{2}a^{-2}-13z^{2}a^{-4}+4z^{2}a^{-6}+3z^{2}a^{-8}-5z^{2}+4za^{-1}+5za^{-3}+za^{-5}+za^{-7}+za^{-9}-a^{2}+3a^{-2}+a^{-4}-2a^{-6}-a^{-8}+1$
The A2 invariant	Data:K11a45/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a45/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a45 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["K11a45"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-6t^{2}+22t-31+22t^{-1}-6t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 87, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+10q^{5}-12q^{4}+14q^{3}-14q^{2}+11q-8+5q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-3z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}+a^{2}z^{2}-5z^{2}a^{-2}-z^{2}a^{-4}+3z^{2}a^{-6}+a^{2}-3a^{-2}+a^{-4}+2a^{-6}-a^{-8}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+4z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}+2az^{7}-9z^{7}a^{-1}-18z^{7}a^{-3}-3z^{7}a^{-5}+4z^{7}a^{-7}+a^{2}z^{6}-22z^{6}a^{-2}-19z^{6}a^{-4}-4z^{6}a^{-6}+3z^{6}a^{-8}-9z^{6}-6az^{5}+12z^{5}a^{-1}+25z^{5}a^{-3}+z^{5}a^{-5}-5z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+37z^{4}a^{-2}+28z^{4}a^{-4}-3z^{4}a^{-6}-6z^{4}a^{-8}+8z^{4}+3az^{3}-11z^{3}a^{-1}-14z^{3}a^{-3}-2z^{3}a^{-7}-2z^{3}a^{-9}+4a^{2}z^{2}-23z^{2}a^{-2}-13z^{2}a^{-4}+4z^{2}a^{-6}+3z^{2}a^{-8}-5z^{2}+4za^{-1}+5za^{-3}+za^{-5}+za^{-7}+za^{-9}-a^{2}+3a^{-2}+a^{-4}-2a^{-6}-a^{-8}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a118,}

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["K11a45"];

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]=

{ $-6t^{2}+22t-31+22t^{-1}-6t^{-2}$, $-q^{8}+3q^{7}-6q^{6}+10q^{5}-12q^{4}+14q^{3}-14q^{2}+11q-8+5q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11a118,}

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 $-8$ $8$ $32$ ${\frac {452}{3}}$ ${\frac {196}{3}}$ $-64$ ${\frac {560}{3}}$ ${\frac {128}{3}}$ $72$ $-{\frac {256}{3}}$ $32$ $-{\frac {3616}{3}}$ $-{\frac {1568}{3}}$ $-{\frac {13951}{15}}$ ${\frac {5284}{15}}$ $-{\frac {51844}{45}}$ ${\frac {1855}{9}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{4591}{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 K11a45. 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                   4 1   -3 11                 6 2     4 9               6 4       -2 7             8 6         2 5           6 6           0 3         5 8             -3 1       4 7               3 -1     1 4                 -3 -3   1 4                   3 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a44 K11a46