K11a17

K11a16

K11a18

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a17 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a17/ThurstonBennequinNumber Hyperbolic Volume 16.1013 A-Polynomial See Data:K11a17/A-polynomial

[edit Notes for K11a17'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 K11a17's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-10t^{2}+30t-41+30t^{-1}-10t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-4z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 123, 2 }
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+13q^{5}-17q^{4}+20q^{3}-20q^{2}+17q-13+8q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-3z^{2}a^{-4}+3z^{2}a^{-6}-2z^{2}+a^{2}-a^{-2}+2a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+8z^{9}a^{-3}+4z^{9}a^{-5}+13z^{8}a^{-2}+14z^{8}a^{-4}+6z^{8}a^{-6}+5z^{8}+3az^{7}-z^{7}a^{-1}-5z^{7}a^{-3}+4z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-34z^{6}a^{-2}-32z^{6}a^{-4}-6z^{6}a^{-6}+3z^{6}a^{-8}-10z^{6}-7az^{5}-11z^{5}a^{-1}-13z^{5}a^{-3}-16z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+28z^{4}a^{-2}+26z^{4}a^{-4}+z^{4}a^{-6}-5z^{4}a^{-8}+5z^{4}+5az^{3}+6z^{3}a^{-1}+10z^{3}a^{-3}+13z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+3a^{2}z^{2}-12z^{2}a^{-2}-8z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-az-2za^{-5}+za^{-9}-a^{2}+a^{-2}-2a^{-6}-a^{-8}$
The A2 invariant	Data:K11a17/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a17/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a17 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["K11a17"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-10t^{2}+30t-41+30t^{-1}-10t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 123, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-7q^{6}+13q^{5}-17q^{4}+20q^{3}-20q^{2}+17q-13+8q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+8z^{9}a^{-3}+4z^{9}a^{-5}+13z^{8}a^{-2}+14z^{8}a^{-4}+6z^{8}a^{-6}+5z^{8}+3az^{7}-z^{7}a^{-1}-5z^{7}a^{-3}+4z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-34z^{6}a^{-2}-32z^{6}a^{-4}-6z^{6}a^{-6}+3z^{6}a^{-8}-10z^{6}-7az^{5}-11z^{5}a^{-1}-13z^{5}a^{-3}-16z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+28z^{4}a^{-2}+26z^{4}a^{-4}+z^{4}a^{-6}-5z^{4}a^{-8}+5z^{4}+5az^{3}+6z^{3}a^{-1}+10z^{3}a^{-3}+13z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+3a^{2}z^{2}-12z^{2}a^{-2}-8z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-az-2za^{-5}+za^{-9}-a^{2}+a^{-2}-2a^{-6}-a^{-8}$

"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["K11a17"];

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]=

{ $t^{3}-10t^{2}+30t-41+30t^{-1}-10t^{-2}+t^{-3}$, $-q^{8}+3q^{7}-7q^{6}+13q^{5}-17q^{4}+20q^{3}-20q^{2}+17q-13+8q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{}

Vassiliev invariants

V2 and V3:

(-1, 2)

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 $-4$ $16$ $8$ ${\frac {322}{3}}$ ${\frac {110}{3}}$ $-64$ ${\frac {448}{3}}$ ${\frac {64}{3}}$ $48$ $-{\frac {32}{3}}$ $128$ $-{\frac {1288}{3}}$ $-{\frac {440}{3}}$ ${\frac {3089}{30}}$ $-{\frac {418}{15}}$ $-{\frac {2222}{45}}$ ${\frac {1615}{18}}$ $-{\frac {1231}{30}}$

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 K11a17. 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                   5 1   -4 11                 8 2     6 9               9 5       -4 7             11 8         3 5           9 9           0 3         8 11             -3 1       6 10               4 -1     2 7                 -5 -3   1 6                   5 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a16 K11a18