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

K11n106 K11n108

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X5,15,6,14 X7,16,8,17 X2,10,3,9 X11,21,12,20 X13,1,14,22 X15,18,16,19 X17,8,18,9 X19,7,20,6 X21,13,22,12

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 + 3z2 + 6z4 + 5z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{21, 4}

Jones Polynomial:

- q-1 + 2 - 2q + 4q2 - 3q3 + 3q4 - 3q5 + 2q6 - q7

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

{...}

A2 (sl(3)) Invariant:

- q-2 + q4 + 2q6 + 2q8 + 2q10 - q12 + q14 - q16 - q20 - q22 - q26 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 5a-6 - 10a-6z2 - 6a-6z4 - a-6z6 + 6a-4 + 18a-4z2 + 17a-4z4 + 7a-4z6 + a-4z8 - a-2 - 6a-2z2 - 5a-2z4 - a-2z6

Kauffman Polynomial:

a-8 - 3a-7z + 7a-7z3 - 5a-7z5 + a-7z7 + 5a-6 - 15a-6z2 + 19a-6z4 - 11a-6z6 + 2a-6z8 - 4a-5z + 9a-5z3 - 3a-5z5 - 3a-5z7 + a-5z9 + 6a-4 - 26a-4z2 + 37a-4z4 - 22a-4z6 + 4a-4z8 - 2a-3z + 8a-3z3 - 3a-3z5 - 3a-3z7 + a-3z9 + a-2 - 11a-2z2 + 18a-2z4 - 11a-2z6 + 2a-2z8 - a-1z + 6a-1z3 - 5a-1z5 + a-1z7

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

{3, 4}

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 11107. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 1 j = 13               1   j = 11             2 1   j = 9           1 1     j = 7         2 2       j = 5       2 1         j = 3     1 3           j = 1   1 1             j = -1   1               j = -3 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, 107]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 107]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 15, 6, 14], X[7, 16, 8, 17], > X[2, 10, 3, 9], X[11, 21, 12, 20], X[13, 1, 14, 22], X[15, 18, 16, 19], > X[17, 8, 18, 9], X[19, 7, 20, 6], X[21, 13, 22, 12]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 107]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, 9, 5, -2, -6, 11, -7, 3, -8, 4, -9, 8, -10, > 6, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 107]]
Out[5]=	DTCode[4, 10, -14, -16, 2, -20, -22, -18, -8, -6, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 107]][t]
Out[6]=	-4 3 4 2 2 3 4 1 + t - -- + -- - - - 2 t + 4 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 107]][z]
Out[7]=	2 4 6 8 1 + 3 z + 6 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 107]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 107]], KnotSignature[Knot[11, NonAlternating, 107]]}
Out[9]=	{21, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 107]][q]
Out[10]=	1 2 3 4 5 6 7 2 - - - 2 q + 4 q - 3 q + 3 q - 3 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 107]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 107]][q]
Out[12]=	-2 4 6 8 10 12 14 16 20 22 26 28 -q + q + 2 q + 2 q + 2 q - q + q - q - q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 107]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 5 6 -2 z 10 z 18 z 6 z 6 z 17 z 5 z z a - -- + -- - a + -- - ----- + ----- - ---- - ---- + ----- - ---- - -- + 6 4 8 6 4 2 6 4 2 6 a a a a a a a a a a 6 6 8 7 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 107]][a, z]
Out[14]=	2 2 2 3 -8 5 6 -2 3 z 4 z 2 z z 15 z 26 z 11 z 7 z a + -- + -- + a - --- - --- - --- - - - ----- - ----- - ----- + ---- + 6 4 7 5 3 a 6 4 2 7 a a a a a a a a a 3 3 3 4 4 4 5 5 5 5 9 z 8 z 6 z 19 z 37 z 18 z 5 z 3 z 3 z 5 z > ---- + ---- + ---- + ----- + ----- + ----- - ---- - ---- - ---- - ---- - 5 3 a 6 4 2 7 5 3 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 9 9 11 z 22 z 11 z z 3 z 3 z z 2 z 4 z 2 z z z > ----- - ----- - ----- + -- - ---- - ---- + -- + ---- + ---- + ---- + -- + -- 6 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 107]], Vassiliev[3][Knot[11, NonAlternating, 107]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 107]][q, t]
Out[16]=	3 3 5 1 1 q q q 5 7 7 2 9 2 3 q + 2 q + ----- + ---- + -- + - + -- + q t + 2 q t + 2 q t + q t + 3 3 2 2 t t q t q t t 9 3 11 3 11 4 13 4 15 5 > q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n107

K11n106 K11n108