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

K11a73 K11a75

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 18 2 6 20 22 8 16

Alexander Polynomial:

t-4 - 5t-3 + 10t-2 - 13t-1 + 15 - 13t + 10t2 - 5t3 + t4

Conway Polynomial:

1 - 2z2 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{73, 4}

Jones Polynomial:

q-2 - 2q-1 + 5 - 7q + 9q2 - 11q3 + 11q4 - 10q5 + 8q6 - 5q7 + 3q8 - q9

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

{...}

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 2 - q2 - 2q6 - 2q8 + q10 - 2q12 + 3q14 + q18 + q20 - q22 + q24 - q26

HOMFLY-PT Polynomial:

- a-6 - 4a-6z2 - 4a-6z4 - a-6z6 + 5a-4 + 13a-4z2 + 13a-4z4 + 6a-4z6 + a-4z8 - 7a-2 - 15a-2z2 - 10a-2z4 - 2a-2z6 + 4 + 4z2 + z4

Kauffman Polynomial:

a-11z3 - a-10z2 + 3a-10z4 - 3a-9z3 + 5a-9z5 + 2a-8z2 - 9a-8z4 + 7a-8z6 - 3a-7z + 8a-7z3 - 16a-7z5 + 8a-7z7 + a-6 + 5a-6z4 - 16a-6z6 + 7a-6z8 - 9a-5z + 26a-5z3 - 18a-5z5 - 5a-5z7 + 4a-5z9 + 5a-4 - 23a-4z2 + 51a-4z4 - 40a-4z6 + 7a-4z8 + a-4z10 - 7a-3z + 8a-3z3 + 18a-3z5 - 23a-3z7 + 6a-3z9 + 7a-2 - 32a-2z2 + 47a-2z4 - 23a-2z6 + a-2z8 + a-2z10 - a-1z - 6a-1z3 + 15a-1z5 - 10a-1z7 + 2a-1z9 + 4 - 12z2 + 13z4 - 6z6 + z8

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

{-2, -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=4 is the signature of 1174. 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 = 19                       1 j = 17                     2   j = 15                   3 1   j = 13                 5 2     j = 11               5 3       j = 9             6 5         j = 7           5 5           j = 5         4 6             j = 3       4 6               j = 1     1 3                 j = -1   1 4                   j = -3   1                     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, 74]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 74]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 8, 15, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[6, 14, 7, 13], X[20, 16, 21, 15], > X[22, 18, 1, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 74]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 74]]
Out[5]=	DTCode[4, 10, 12, 14, 18, 2, 6, 20, 22, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 74]][t]
Out[6]=	-4 5 10 13 2 3 4 15 + t - -- + -- - -- - 13 t + 10 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 74]][z]
Out[7]=	2 6 8 1 - 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 74]}
In[9]:=	{KnotDet[Knot[11, Alternating, 74]], KnotSignature[Knot[11, Alternating, 74]]}
Out[9]=	{73, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 74]][q]
Out[10]=	-2 2 2 3 4 5 6 7 8 9 5 + q - - - 7 q + 9 q - 11 q + 11 q - 10 q + 8 q - 5 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 74]}
In[12]:=	A2Invariant[Knot[11, Alternating, 74]][q]
Out[12]=	-6 -4 -2 2 6 8 10 12 14 18 20 2 + q + q + q - q - 2 q - 2 q + q - 2 q + 3 q + q + q - 22 24 26 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 74]][a, z]
Out[13]=	2 2 2 4 4 4 -6 5 7 2 4 z 13 z 15 z 4 4 z 13 z 10 z 4 - a + -- - -- + 4 z - ---- + ----- - ----- + z - ---- + ----- - ----- - 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 6 8 z 6 z 2 z z > -- + ---- - ---- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 74]][a, z]
Out[14]=	2 2 2 2 -6 5 7 3 z 9 z 7 z z 2 z 2 z 23 z 32 z 4 + a + -- + -- - --- - --- - --- - - - 12 z - --- + ---- - ----- - ----- + 4 2 7 5 3 a 10 8 4 2 a a a a a a a a a 3 3 3 3 3 3 4 4 4 z 3 z 8 z 26 z 8 z 6 z 4 3 z 9 z 5 z > --- - ---- + ---- + ----- + ---- - ---- + 13 z + ---- - ---- + ---- + 11 9 7 5 3 a 10 8 6 a a a a a a a a 4 4 5 5 5 5 5 6 51 z 47 z 5 z 16 z 18 z 18 z 15 z 6 7 z > ----- + ----- + ---- - ----- - ----- + ----- + ----- - 6 z + ---- - 4 2 9 7 5 3 a 8 a a a a a a a 6 6 6 7 7 7 7 8 8 16 z 40 z 23 z 8 z 5 z 23 z 10 z 8 7 z 7 z > ----- - ----- - ----- + ---- - ---- - ----- - ----- + z + ---- + ---- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 8 9 9 9 10 10 z 4 z 6 z 2 z z z > -- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 74]], Vassiliev[3][Knot[11, Alternating, 74]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, Alternating, 74]][q, t]
Out[16]=	3 3 5 1 1 1 4 q 3 q 4 q 5 7 6 q + 4 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 6 q t + 5 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 5 q t + 6 q t + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 15 5 15 6 17 6 19 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a74

K11a73 K11a75