K11n15

K11n14

K11n16

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n15 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X11,19,12,18 X13,21,14,20 X15,6,16,7 X17,1,18,22 X19,13,20,12 X21,15,22,14
Gauss code	1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 10 -16 2 -18 -20 -6 -22 -12 -14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n15/ThurstonBennequinNumber Hyperbolic Volume 9.94805 A-Polynomial See Data:K11n15/A-polynomial

[edit Notes for K11n15'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 K11n15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3}$
Conway polynomial	$z^6+2 z^4+z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, 2 }
Jones polynomial	$-q^8+2 q^7-3 q^6+5 q^5-6 q^4+6 q^3-5 q^2+4 q-2+ q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^6 a^{-4} -2 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} -6 z^2 a^{-2} +9 z^2 a^{-4} -3 z^2 a^{-6} +z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2$
Kauffman polynomial (db, data sources)	$z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} -5 z^7 a^{-3} -3 z^7 a^{-5} +2 z^7 a^{-7} -5 z^6 a^{-2} -14 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} +12 z^5 a^{-3} +4 z^5 a^{-5} -5 z^5 a^{-7} +z^5 a^{-9} +14 z^4 a^{-2} +29 z^4 a^{-4} +10 z^4 a^{-6} -6 z^4 a^{-8} +z^4-4 z^3 a^{-1} -8 z^3 a^{-3} +z^3 a^{-5} +2 z^3 a^{-7} -3 z^3 a^{-9} -15 z^2 a^{-2} -21 z^2 a^{-4} -6 z^2 a^{-6} +3 z^2 a^{-8} -3 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-9} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2$
The A2 invariant	$q^4+q^2+ q^{-2} -2 q^{-4} +2 q^{-12} +2 q^{-16} - q^{-20} - q^{-24}$
The G2 invariant	Data:K11n15/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n15 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["K11n15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^6+2 z^4+z^2+1$

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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^6 a^{-4} -2 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} -6 z^2 a^{-2} +9 z^2 a^{-4} -3 z^2 a^{-6} +z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} -5 z^7 a^{-3} -3 z^7 a^{-5} +2 z^7 a^{-7} -5 z^6 a^{-2} -14 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} +12 z^5 a^{-3} +4 z^5 a^{-5} -5 z^5 a^{-7} +z^5 a^{-9} +14 z^4 a^{-2} +29 z^4 a^{-4} +10 z^4 a^{-6} -6 z^4 a^{-8} +z^4-4 z^3 a^{-1} -8 z^3 a^{-3} +z^3 a^{-5} +2 z^3 a^{-7} -3 z^3 a^{-9} -15 z^2 a^{-2} -21 z^2 a^{-4} -6 z^2 a^{-6} +3 z^2 a^{-8} -3 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-9} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n56, K11n58,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_12,}

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["K11n15"];

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]=

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

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]=

{8_16, 10_156, K11n56, K11n58,}

In[6]:=

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

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

Out[6]=

{9_12,}

Vassiliev invariants

V2 and V3:

(1, 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 $4$ $24$ $8$ $\frac{158}{3}$ $-\frac{38}{3}$ $96$ $144$ $-32$ $-8$ $\frac{32}{3}$ $288$ $\frac{632}{3}$ $-\frac{152}{3}$ $\frac{20911}{30}$ $\frac{286}{5}$ $\frac{422}{45}$ $-\frac{175}{18}$ $-\frac{689}{30}$

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 $t^rq^j$ 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 K11n15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11n14 K11n16