© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a118 K11a120

Knotscape

This page is passe. Go here instead!    The Knot K11a119 Visit K11a119's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X12,10,13,9 X2,11,3,12 X8,14,9,13 X22,16,1,15 X20,18,21,17 X6,19,7,20 X16,22,17,21

Gauss Code:

{1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, -9, 11, -8}

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 12 2 8 22 20 6 16

Alexander Polynomial:

4t-2 - 19t-1 + 31 - 19t + 4t2

Conway Polynomial:

1 - 3z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a98, ...}

Determinant and Signature:

{77, 0}

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 8q-1 + 10 - 12q + 12q2 - 10q3 + 8q4 - 5q5 + 3q6 - q7

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + q-6 - 3q-4 + q-2 - 2 - q2 + q4 + 3q8 - q10 + 2q14 - 2q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4z4 + 2a-2 + 2a-2z2 + 2a-2z4 - 2 - 2z2 + z4 - 2a2z2 + a4

Kauffman Polynomial:

4a-7z3 - 4a-7z5 + a-7z7 - 5a-6z2 + 16a-6z4 - 13a-6z6 + 3a-6z8 + a-5z - 4a-5z3 + 11a-5z5 - 11a-5z7 + 3a-5z9 - 5a-4z2 + 15a-4z4 - 15a-4z6 + 2a-4z8 + a-4z10 + a-3z - 6a-3z3 + 11a-3z5 - 14a-3z7 + 5a-3z9 - 2a-2 + 8a-2z2 - 6a-2z4 - 5a-2z6 + 2a-2z8 + a-2z10 - 4a-1z + 12a-1z3 - 12a-1z5 + 2a-1z7 + 2a-1z9 - 2 + 10z2 - 8z4 + 3z8 - 4az + 8az3 - 6az5 + 4az7 - 2a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-3, 1}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 5 2     j = 7               5 3       j = 5             7 5         j = 3           5 5           j = 1         5 7             j = -1       4 6               j = -3     1 4                 j = -5   1 4                   j = -7   1                     j = -9 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 119]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 119]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 119]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 119]]
Out[5]=	DTCode[4, 10, 14, 18, 12, 2, 8, 22, 20, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 119]][t]
Out[6]=	4 19 2 31 + -- - -- - 19 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 119]][z]
Out[7]=	2 4 1 - 3 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 98], Knot[11, Alternating, 119]}
In[9]:=	{KnotDet[Knot[11, Alternating, 119]], KnotSignature[Knot[11, Alternating, 119]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 119]][q]
Out[10]=	-4 2 5 8 2 3 4 5 6 7 10 + q - -- + -- - - - 12 q + 12 q - 10 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 119]}
In[12]:=	A2Invariant[Knot[11, Alternating, 119]][q]
Out[12]=	-14 -12 -10 2 -6 3 -2 2 4 8 10 14 -2 + q + q - q + -- + q - -- + q - q + q + 3 q - q + 2 q - 8 4 q q 16 18 20 22 > 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 119]][a, z]
Out[13]=	2 2 4 4 2 4 2 z 2 z 2 2 4 z 2 z -2 + -- + a - 2 z - -- + ---- - 2 a z + z + -- + ---- 2 6 2 4 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 119]][a, z]
Out[14]=	2 2 2 2 4 z z 4 z 2 5 z 5 z 8 z 4 2 -2 - -- + a + -- + -- - --- - 4 a z + 10 z - ---- - ---- + ---- - 2 a z + 2 5 3 a 6 4 2 a a a a a a 3 3 3 3 4 4 4 z 4 z 6 z 12 z 3 3 3 4 16 z 15 z > ---- - ---- - ---- + ----- + 8 a z - 2 a z - 8 z + ----- + ----- - 7 5 3 a 6 4 a a a a a 4 5 5 5 5 6 z 2 4 4 4 4 z 11 z 11 z 12 z 5 3 5 > ---- - 2 a z + a z - ---- + ----- + ----- - ----- - 6 a z + 2 a z - 2 7 5 3 a a a a a 6 6 6 7 7 7 7 13 z 15 z 5 z 2 6 z 11 z 14 z 2 z 7 > ----- - ----- - ---- + 3 a z + -- - ----- - ----- + ---- + 4 a z + 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 3 z 2 z 2 z 3 z 5 z 2 z z z > 3 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 119]], Vassiliev[3][Knot[11, Alternating, 119]]}
Out[15]=	{-3, 1}
In[16]:=	Kh[Knot[11, Alternating, 119]][q, t]
Out[16]=	6 1 1 1 4 1 4 4 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 5 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 5 q t + 7 q t + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a119

K11a118 K11a120