K11n70

K11n69

K11n71

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n70 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X2837 X9,19,10,18 X11,21,12,20 X13,6,14,7 X15,22,16,1 X17,13,18,12 X19,11,20,10 X21,16,22,17
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 9, -7, 3, -8, 11, -9, 5, -10, 6, -11, 8
Dowker-Thistlethwaite code	4 8 -14 2 -18 -20 -6 -22 -12 -10 -16

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n70/ThurstonBennequinNumber Hyperbolic Volume 6.72199 A-Polynomial See Data:K11n70/A-polynomial

[edit Notes for K11n70's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11n70's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+2t^{2}-2t+3-2t^{-1}+2t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-4z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 4 }
Jones polynomial	$q^{6}-q^{5}+q^{4}-2q^{3}+2q^{2}-2q+2-q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-6z^{4}a^{-2}+z^{4}a^{-4}+z^{4}-11z^{2}a^{-2}+4z^{2}a^{-4}+4z^{2}-6a^{-2}+3a^{-4}+4$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+3z^{8}a^{-2}+2z^{8}a^{-4}+z^{8}-6z^{7}a^{-1}-4z^{7}a^{-3}+2z^{7}a^{-5}-19z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+10z^{5}a^{-1}-z^{5}a^{-3}-11z^{5}a^{-5}+37z^{4}a^{-2}+16z^{4}a^{-4}-5z^{4}a^{-6}+16z^{4}-4z^{3}a^{-1}+11z^{3}a^{-3}+15z^{3}a^{-5}-27z^{2}a^{-2}-9z^{2}a^{-4}+4z^{2}a^{-6}-14z^{2}-za^{-1}-6za^{-3}-6za^{-5}-za^{-7}+6a^{-2}+3a^{-4}+4$
The A2 invariant	Data:K11n70/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n70/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n70 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["K11n70"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+2t^{2}-2t+3-2t^{-1}+2t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 13, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-q^{5}+q^{4}-2q^{3}+2q^{2}-2q+2-q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-6z^{4}a^{-2}+z^{4}a^{-4}+z^{4}-11z^{2}a^{-2}+4z^{2}a^{-4}+4z^{2}-6a^{-2}+3a^{-4}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+3z^{8}a^{-2}+2z^{8}a^{-4}+z^{8}-6z^{7}a^{-1}-4z^{7}a^{-3}+2z^{7}a^{-5}-19z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+10z^{5}a^{-1}-z^{5}a^{-3}-11z^{5}a^{-5}+37z^{4}a^{-2}+16z^{4}a^{-4}-5z^{4}a^{-6}+16z^{4}-4z^{3}a^{-1}+11z^{3}a^{-3}+15z^{3}a^{-5}-27z^{2}a^{-2}-9z^{2}a^{-4}+4z^{2}a^{-6}-14z^{2}-za^{-1}-6za^{-3}-6za^{-5}-za^{-7}+6a^{-2}+3a^{-4}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {8_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["K11n70"];

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]=

{ $-t^{3}+2t^{2}-2t+3-2t^{-1}+2t^{-2}-t^{-3}$, $q^{6}-q^{5}+q^{4}-2q^{3}+2q^{2}-2q+2-q^{-1}+q^{-2}$ }

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]=

{8_1,}

Vassiliev invariants

V2 and V3:

(-3, -3)

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 $-12$ $-24$ $72$ $114$ $70$ $288$ $528$ $160$ $104$ $-288$ $288$ $-1368$ $-840$ $-{\frac {5471}{10}}$ ${\frac {11186}{15}}$ $-{\frac {8114}{5}}$ ${\frac {597}{2}}$ $-{\frac {4511}{10}}$

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 K11n70. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 13                 1 1 11                   0 9             2 2   0 7           1       -1 5         1 2 1     0 3       2 2         0 1       1 1         0 -1   1 2             1 -3                   0 -5 1                 1

Integral Khovanov Homology (db, data source)

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

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.

K11n69 K11n71