K11a348

K11a347

K11a349

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a348 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X18,4,19,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X2,11,3,12 X20,14,21,13 X22,16,1,15 X10,18,11,17 X8,19,9,20 X14,22,15,21
Gauss code	1, -6, 2, -5, 3, -1, 4, -10, 5, -9, 6, -4, 7, -11, 8, -3, 9, -2, 10, -7, 11, -8
Dowker-Thistlethwaite code	6 18 16 12 4 2 20 22 10 8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a348/ThurstonBennequinNumber Hyperbolic Volume 17.56 A-Polynomial See Data:K11a348/A-polynomial

[edit Notes for K11a348's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a348's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+19t^{2}-29t+33-29t^{-1}+19t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 145, 4 }
Jones polynomial	$q^{10}-4q^{9}+9q^{8}-15q^{7}+20q^{6}-23q^{5}+23q^{4}-20q^{3}+15q^{2}-9q+5-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+4z^{6}a^{-4}-2z^{6}a^{-6}-2z^{4}a^{-2}+4z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-2}-z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}+3a^{-2}-3a^{-4}+a^{-6}$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-4}+4z^{10}a^{-6}+8z^{9}a^{-3}+19z^{9}a^{-5}+11z^{9}a^{-7}+5z^{8}a^{-2}+z^{8}a^{-4}+11z^{8}a^{-6}+15z^{8}a^{-8}+z^{7}a^{-1}-27z^{7}a^{-3}-56z^{7}a^{-5}-14z^{7}a^{-7}+14z^{7}a^{-9}-16z^{6}a^{-2}-33z^{6}a^{-4}-50z^{6}a^{-6}-24z^{6}a^{-8}+9z^{6}a^{-10}-2z^{5}a^{-1}+26z^{5}a^{-3}+46z^{5}a^{-5}-5z^{5}a^{-7}-19z^{5}a^{-9}+4z^{5}a^{-11}+12z^{4}a^{-2}+34z^{4}a^{-4}+41z^{4}a^{-6}+11z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}-7z^{3}a^{-3}-12z^{3}a^{-5}+5z^{3}a^{-7}+9z^{3}a^{-9}-z^{3}a^{-11}+2z^{2}a^{-2}-2z^{2}a^{-4}-9z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}+za^{-3}+za^{-5}-za^{-7}-za^{-9}-3a^{-2}-3a^{-4}-a^{-6}$
The A2 invariant	$-q^{2}+3-q^{-2}+3q^{-4}+2q^{-6}-3q^{-8}+4q^{-10}-6q^{-12}+2q^{-14}-q^{-16}-q^{-18}+4q^{-20}-3q^{-22}+2q^{-24}-q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	Data:K11a348/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a348 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["K11a348"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-7t^{3}+19t^{2}-29t+33-29t^{-1}+19t^{-2}-7t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 145, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-4q^{9}+9q^{8}-15q^{7}+20q^{6}-23q^{5}+23q^{4}-20q^{3}+15q^{2}-9q+5-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-4}+4z^{10}a^{-6}+8z^{9}a^{-3}+19z^{9}a^{-5}+11z^{9}a^{-7}+5z^{8}a^{-2}+z^{8}a^{-4}+11z^{8}a^{-6}+15z^{8}a^{-8}+z^{7}a^{-1}-27z^{7}a^{-3}-56z^{7}a^{-5}-14z^{7}a^{-7}+14z^{7}a^{-9}-16z^{6}a^{-2}-33z^{6}a^{-4}-50z^{6}a^{-6}-24z^{6}a^{-8}+9z^{6}a^{-10}-2z^{5}a^{-1}+26z^{5}a^{-3}+46z^{5}a^{-5}-5z^{5}a^{-7}-19z^{5}a^{-9}+4z^{5}a^{-11}+12z^{4}a^{-2}+34z^{4}a^{-4}+41z^{4}a^{-6}+11z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}-7z^{3}a^{-3}-12z^{3}a^{-5}+5z^{3}a^{-7}+9z^{3}a^{-9}-z^{3}a^{-11}+2z^{2}a^{-2}-2z^{2}a^{-4}-9z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}+za^{-3}+za^{-5}-za^{-7}-za^{-9}-3a^{-2}-3a^{-4}-a^{-6}$

"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["K11a348"];

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]=

{ $t^{4}-7t^{3}+19t^{2}-29t+33-29t^{-1}+19t^{-2}-7t^{-3}+t^{-4}$, $q^{10}-4q^{9}+9q^{8}-15q^{7}+20q^{6}-23q^{5}+23q^{4}-20q^{3}+15q^{2}-9q+5-q^{-1}$ }

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:

(0, -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 $0$ $-8$ $0$ $0$ $24$ $0$ ${\frac {304}{3}}$ ${\frac {352}{3}}$ $24$ $0$ $32$ $0$ $0$ $432$ $312$ $152$ $-16$ $0$

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=$4 is the signature of K11a348. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   6 1   5 15                 9 3     -6 13               11 6       5 11             12 9         -3 9           11 11           0 7         9 12             3 5       6 11               -5 3     4 10                 6 1   1 5                   -4 -1   4                     4 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a347 K11a349