K11a248

K11a247

K11a249

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a248 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X20,11,21,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22
Gauss code	1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code	6 8 10 18 16 20 22 4 2 14 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a248/ThurstonBennequinNumber Hyperbolic Volume 17.7145 A-Polynomial See Data:K11a248/A-polynomial

[edit Notes for K11a248's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a248's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 159, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-15+22q^{-1}-25q^{-2}+26q^{-3}-23q^{-4}+17q^{-5}-11q^{-6}+5q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-a^{6}z^{2}+5a^{4}z^{2}-7a^{2}z^{2}+3z^{2}+1$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+17a^{3}z^{9}+8az^{9}+13a^{6}z^{8}+17a^{4}z^{8}+12a^{2}z^{8}+8z^{8}+11a^{7}z^{7}-2a^{5}z^{7}-33a^{3}z^{7}-16az^{7}+4z^{7}a^{-1}+5a^{8}z^{6}-17a^{6}z^{6}-44a^{4}z^{6}-44a^{2}z^{6}+z^{6}a^{-2}-21z^{6}+a^{9}z^{5}-15a^{7}z^{5}-14a^{5}z^{5}+21a^{3}z^{5}+10az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+4a^{6}z^{4}+32a^{4}z^{4}+45a^{2}z^{4}-2z^{4}a^{-2}+19z^{4}+4a^{7}z^{3}+3a^{5}z^{3}-7a^{3}z^{3}-2az^{3}+4z^{3}a^{-1}-a^{6}z^{2}-9a^{4}z^{2}-15a^{2}z^{2}-7z^{2}+a^{7}z+3a^{5}z+3a^{3}z+az+1$
The A2 invariant	$-q^{24}+2q^{22}-2q^{18}+4q^{16}-5q^{14}+q^{12}-2q^{8}+6q^{6}-4q^{4}+5q^{2}-1-2q^{-2}+3q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	Data:K11a248/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a248 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["K11a248"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 159, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-15+22q^{-1}-25q^{-2}+26q^{-3}-23q^{-4}+17q^{-5}-11q^{-6}+5q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-a^{6}z^{2}+5a^{4}z^{2}-7a^{2}z^{2}+3z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+17a^{3}z^{9}+8az^{9}+13a^{6}z^{8}+17a^{4}z^{8}+12a^{2}z^{8}+8z^{8}+11a^{7}z^{7}-2a^{5}z^{7}-33a^{3}z^{7}-16az^{7}+4z^{7}a^{-1}+5a^{8}z^{6}-17a^{6}z^{6}-44a^{4}z^{6}-44a^{2}z^{6}+z^{6}a^{-2}-21z^{6}+a^{9}z^{5}-15a^{7}z^{5}-14a^{5}z^{5}+21a^{3}z^{5}+10az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+4a^{6}z^{4}+32a^{4}z^{4}+45a^{2}z^{4}-2z^{4}a^{-2}+19z^{4}+4a^{7}z^{3}+3a^{5}z^{3}-7a^{3}z^{3}-2az^{3}+4z^{3}a^{-1}-a^{6}z^{2}-9a^{4}z^{2}-15a^{2}z^{2}-7z^{2}+a^{7}z+3a^{5}z+3a^{3}z+az+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a71,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a71,}

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["K11a248"];

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]=

{ $-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+9q-15+22q^{-1}-25q^{-2}+26q^{-3}-23q^{-4}+17q^{-5}-11q^{-6}+5q^{-7}-q^{-8}$ }

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]=

{K11a71,}

In[6]:=

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

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

Out[6]=

{K11a71,}

Vassiliev invariants

V2 and V3:

(0, 0)

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 $0$ $0$ $0$ $16$ $16$ $0$ $-64$ $32$ $-96$ $0$ $0$ $0$ $0$ $136$ $-{\frac {832}{3}}$ ${\frac {896}{3}}$ ${\frac {344}{3}}$ $40$

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 K11a248. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 9 3     -6 -1               13 6       7 -3             13 10         -3 -5           13 12           1 -7         10 13             3 -9       7 13               -6 -11     4 10                 6 -13   1 7                   -6 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a247 K11a249