K11a247

K11a246

K11a248

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a247 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a247/ThurstonBennequinNumber Hyperbolic Volume 3.55382 A-Polynomial See Data:K11a247/A-polynomial

[edit Notes for K11a247's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for K11a247's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t-9+5t^{-1}$
Conway polynomial	$5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, 2 }
Jones polynomial	$-q^{12}+q^{11}-q^{10}+2q^{9}-2q^{8}+2q^{7}-2q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{-2}+z^{2}a^{-4}+z^{2}a^{-6}+z^{2}a^{-8}+z^{2}a^{-10}+a^{-2}+a^{-10}-a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+2z^{9}a^{-11}+z^{9}a^{-13}+z^{8}a^{-8}-7z^{8}a^{-10}-8z^{8}a^{-12}+z^{7}a^{-7}-5z^{7}a^{-9}-14z^{7}a^{-11}-8z^{7}a^{-13}+z^{6}a^{-6}-4z^{6}a^{-8}+17z^{6}a^{-10}+22z^{6}a^{-12}+z^{5}a^{-5}-3z^{5}a^{-7}+6z^{5}a^{-9}+31z^{5}a^{-11}+21z^{5}a^{-13}+z^{4}a^{-4}-2z^{4}a^{-6}+3z^{4}a^{-8}-19z^{4}a^{-10}-25z^{4}a^{-12}+z^{3}a^{-3}-z^{3}a^{-5}+z^{3}a^{-7}-z^{3}a^{-9}-24z^{3}a^{-11}-20z^{3}a^{-13}+z^{2}a^{-2}+10z^{2}a^{-10}+11z^{2}a^{-12}+5za^{-11}+5za^{-13}-a^{-2}-a^{-10}-a^{-12}$
The A2 invariant	Data:K11a247/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a247/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a247 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["K11a247"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$5t-9+5t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$5z^{2}+1$

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]=

{ 19, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{12}+q^{11}-q^{10}+2q^{9}-2q^{8}+2q^{7}-2q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+q$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{-2}+z^{2}a^{-4}+z^{2}a^{-6}+z^{2}a^{-8}+z^{2}a^{-10}+a^{-2}+a^{-10}-a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+2z^{9}a^{-11}+z^{9}a^{-13}+z^{8}a^{-8}-7z^{8}a^{-10}-8z^{8}a^{-12}+z^{7}a^{-7}-5z^{7}a^{-9}-14z^{7}a^{-11}-8z^{7}a^{-13}+z^{6}a^{-6}-4z^{6}a^{-8}+17z^{6}a^{-10}+22z^{6}a^{-12}+z^{5}a^{-5}-3z^{5}a^{-7}+6z^{5}a^{-9}+31z^{5}a^{-11}+21z^{5}a^{-13}+z^{4}a^{-4}-2z^{4}a^{-6}+3z^{4}a^{-8}-19z^{4}a^{-10}-25z^{4}a^{-12}+z^{3}a^{-3}-z^{3}a^{-5}+z^{3}a^{-7}-z^{3}a^{-9}-24z^{3}a^{-11}-20z^{3}a^{-13}+z^{2}a^{-2}+10z^{2}a^{-10}+11z^{2}a^{-12}+5za^{-11}+5za^{-13}-a^{-2}-a^{-10}-a^{-12}$

"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["K11a247"];

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]=

{ $5t-9+5t^{-1}$, $-q^{12}+q^{11}-q^{10}+2q^{9}-2q^{8}+2q^{7}-2q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+q$ }

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:

(5, 15)

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 $20$ $120$ $200$ ${\frac {2230}{3}}$ ${\frac {290}{3}}$ $2400$ $5200$ $800$ $760$ ${\frac {4000}{3}}$ $7200$ ${\frac {44600}{3}}$ ${\frac {5800}{3}}$ ${\frac {223375}{6}}$ $-{\frac {1142}{3}}$ ${\frac {138182}{9}}$ ${\frac {12565}{18}}$ ${\frac {11791}{6}}$

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 K11a247. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 25                       1 -1 23                         0 21                   1 1   0 19                 1       1 17               1 1       0 15             1 1         0 13           1 1           0 11         1 1             0 9       1 1               0 7     1 1                 0 5     1                   1 3 1 1                     0 1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a246 K11a248