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

K11a196 K11a198

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 14 18 2 8 22 20 10 6

Alexander Polynomial:

3t-3 - 14t-2 + 33t-1 - 43 + 33t - 14t2 + 3t3

Conway Polynomial:

1 + 4z2 + 4z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{143, 2}

Jones Polynomial:

- q-2 + 3q-1 - 7 + 14q - 19q2 + 23q3 - 23q4 + 21q5 - 16q6 + 10q7 - 5q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 2 + 5q2 - 3q4 + 3q6 + 3q8 - 2q10 + 4q12 - 4q14 + 3q16 - 3q20 + 3q22 - 3q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - 2a-6 - 5a-6z2 - 3a-6z4 + 2a-4 + 7a-4z2 + 6a-4z4 + 2a-4z6 + 2a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

- a-10z4 + a-10z6 + a-9z + 4a-9z3 - 10a-9z5 + 5a-9z7 - 2a-8z2 + 11a-8z4 - 20a-8z6 + 9a-8z8 - 3a-7z + 14a-7z3 - 17a-7z5 - 5a-7z7 + 7a-7z9 + 2a-6 - 15a-6z2 + 42a-6z4 - 51a-6z6 + 17a-6z8 + 2a-6z10 - 6a-5z + 16a-5z3 - 6a-5z5 - 18a-5z7 + 13a-5z9 + 2a-4 - 15a-4z2 + 41a-4z4 - 42a-4z6 + 15a-4z8 + 2a-4z10 - 2a-3z + 8a-3z3 - 5a-3z5 - 3a-3z7 + 6a-3z9 - 2a-2 + a-2z2 + 6a-2z4 - 9a-2z6 + 7a-2z8 + a-1z - 5a-1z5 + 5a-1z7 - 1 + 3z2 - 5z4 + 3z6 + az - 2az3 + az5

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

{4, 6}

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=2 is the signature of 11197. 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 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     4   j = 15                   6 1   j = 13                 10 4     j = 11               11 6       j = 9             12 10         j = 7           11 11           j = 5         8 12             j = 3       6 11               j = 1     2 9                 j = -1   1 5                   j = -3   2                     j = -5 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, 197]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 197]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[8, 14, 9, 13], X[22, 15, 1, 16], > X[20, 18, 21, 17], X[10, 20, 11, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 197]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -7, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 197]]
Out[5]=	DTCode[4, 12, 16, 14, 18, 2, 8, 22, 20, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 197]][t]
Out[6]=	3 14 33 2 3 -43 + -- - -- + -- + 33 t - 14 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 197]][z]
Out[7]=	2 4 6 1 + 4 z + 4 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 197]}
In[9]:=	{KnotDet[Knot[11, Alternating, 197]], KnotSignature[Knot[11, Alternating, 197]]}
Out[9]=	{143, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 197]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 9 -7 - q + - + 14 q - 19 q + 23 q - 23 q + 21 q - 16 q + 10 q - 5 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 197]}
In[12]:=	A2Invariant[Knot[11, Alternating, 197]][q]
Out[12]=	-6 -4 -2 2 4 6 8 10 12 14 -2 - q + q - q + 5 q - 3 q + 3 q + 3 q - 2 q + 4 q - 4 q + 16 20 22 24 26 28 > 3 q - 3 q + 3 q - 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 197]][a, z]
Out[13]=	2 2 2 2 4 4 4 2 2 2 2 z 5 z 7 z 3 z 4 3 z 6 z 2 z -1 - -- + -- + -- - 2 z + -- - ---- + ---- + ---- - z - ---- + ---- + ---- + 6 4 2 8 6 4 2 6 4 2 a a a a a a a a a a 6 6 2 z z > ---- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 197]][a, z]
Out[14]=	2 2 2 2 2 z 3 z 6 z 2 z z 2 2 z 15 z -1 + -- + -- - -- + -- - --- - --- - --- + - + a z + 3 z - ---- - ----- - 6 4 2 9 7 5 3 a 8 6 a a a a a a a a a 2 2 3 3 3 3 4 4 15 z z 4 z 14 z 16 z 8 z 3 4 z 11 z > ----- + -- + ---- + ----- + ----- + ---- - 2 a z - 5 z - --- + ----- + 4 2 9 7 5 3 10 8 a a a a a a a a 4 4 4 5 5 5 5 5 42 z 41 z 6 z 10 z 17 z 6 z 5 z 5 z 5 6 > ----- + ----- + ---- - ----- - ----- - ---- - ---- - ---- + a z + 3 z + 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 6 7 7 7 7 7 z 20 z 51 z 42 z 9 z 5 z 5 z 18 z 3 z 5 z > --- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ---- + ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 9 z 17 z 15 z 7 z 7 z 13 z 6 z 2 z 2 z > ---- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 197]], Vassiliev[3][Knot[11, Alternating, 197]]}
Out[15]=	{4, 6}
In[16]:=	Kh[Knot[11, Alternating, 197]][q, t]
Out[16]=	3 1 2 1 5 2 q 3 5 5 2 9 q + 6 q + ----- + ----- + ---- + --- + --- + 11 q t + 8 q t + 12 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 11 q t + 11 q t + 12 q t + 10 q t + 11 q t + 6 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 10 q t + 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a197

K11a196 K11a198