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

K11n8 K11n10

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,19,12,18 X6,14,7,13 X15,21,16,20 X17,1,18,22 X19,13,20,12 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 14 2 -18 6 -20 -22 -12 -16

Alexander Polynomial:

- t-4 + 3t-3 - t-2 - 4t-1 + 7 - 4t - t2 + 3t3 - t4

Conway Polynomial:

1 + 3z2 - 3z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{5, 4}

Jones Polynomial:

q - q2 + 2q3 - q4 + q6 - 2q7 + 2q8 - 2q9 + 2q10 - q11

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

{...}

A2 (sl(3)) Invariant:

q4 + q6 + q8 + 2q10 + q12 - q14 - q16 - q18 - 2q20 + q22 + 2q26 + q32 - 2q34

HOMFLY-PT Polynomial:

- 2a-10 - 2a-10z2 + 6a-8 + 12a-8z2 + 7a-8z4 + a-8z6 - 8a-6 - 17a-6z2 - 16a-6z4 - 7a-6z6 - a-6z8 + 5a-4 + 10a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

a-13z - 3a-13z3 + a-13z5 + 4a-12z2 - 7a-12z4 + 2a-12z6 - a-11z + a-11z3 - 3a-11z5 + a-11z7 + 2a-10 - a-10z2 - 3a-10z4 + a-10z6 - 4a-9z + 11a-9z3 - 7a-9z5 + a-9z7 + 6a-8 - 22a-8z2 + 31a-8z4 - 15a-8z6 + 2a-8z8 - 4a-7z + 5a-7z3 + 6a-7z5 - 6a-7z7 + a-7z9 + 8a-6 - 32a-6z2 + 43a-6z4 - 21a-6z6 + 3a-6z8 - 2a-5z - 2a-5z3 + 9a-5z5 - 6a-5z7 + a-5z9 + 5a-4 - 15a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8

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

{3, 7}

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=4 is the signature of 119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     1   j = 19                   1 1   j = 17               1 2 1     j = 15             1 2 1       j = 13           1 2 2         j = 11         1 2 2           j = 9       1 1 1             j = 7     1 1 1               j = 5   1 2                   j = 3                         j = 1 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, NonAlternating, 9]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 9]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[6, 14, 7, 13], X[15, 21, 16, 20], > X[17, 1, 18, 22], X[19, 13, 20, 12], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 9]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 10, 7, -4, -8, 11, -9, 6, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 9]]
Out[5]=	DTCode[4, 8, 10, 14, 2, -18, 6, -20, -22, -12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 9]][t]
Out[6]=	-4 3 -2 4 2 3 4 7 - t + -- - t - - - 4 t - t + 3 t - t 3 t t
In[7]:=	Conway[Knot[11, NonAlternating, 9]][z]
Out[7]=	2 4 6 8 1 + 3 z - 3 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 9]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 9]], KnotSignature[Knot[11, NonAlternating, 9]]}
Out[9]=	{5, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 9]][q]
Out[10]=	2 3 4 6 7 8 9 10 11 q - q + 2 q - q + q - 2 q + 2 q - 2 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 9]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 9]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 26 32 34 q + q + q + 2 q + q - q - q - q - 2 q + q + 2 q + q - 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 9]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -2 6 8 5 2 z 12 z 17 z 10 z 7 z 16 z 6 z z --- + -- - -- + -- - ---- + ----- - ----- + ----- + ---- - ----- + ---- + -- - 10 8 6 4 10 8 6 4 8 6 4 8 a a a a a a a a a a a a 6 6 8 7 z z z > ---- + -- - -- 6 4 6 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 9]][a, z]
Out[14]=	2 2 2 2 2 6 8 5 z z 4 z 4 z 2 z 4 z z 22 z 32 z --- + -- + -- + -- + --- - --- - --- - --- - --- + ---- - --- - ----- - ----- - 10 8 6 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 15 z 3 z z 11 z 5 z 2 z 7 z 3 z 31 z 43 z > ----- - ---- + --- + ----- + ---- - ---- - ---- - ---- + ----- + ----- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 16 z z 3 z 7 z 6 z 9 z 2 z z 15 z 21 z > ----- + --- - ---- - ---- + ---- + ---- + ---- + --- - ----- - ----- - 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 7 z z z 6 z 6 z 2 z 3 z z z z > ---- + --- + -- - ---- - ---- + ---- + ---- + -- + -- + -- 4 11 9 7 5 8 6 4 7 5 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 9]], Vassiliev[3][Knot[11, NonAlternating, 9]]}
Out[15]=	{3, 7}
In[16]:=	Kh[Knot[11, NonAlternating, 9]][q, t]
Out[16]=	5 5 7 q q 7 9 7 2 9 2 11 2 9 3 11 3 2 q + q + -- + -- + q t + q t + q t + q t + q t + q t + 2 q t + 2 t t 13 3 11 4 13 4 15 4 13 5 15 5 17 5 > q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > q t + 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n9

K11n8 K11n10