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

K11a178 K11a180

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 20 2 6 8 22 10

Alexander Polynomial:

t-4 - 5t-3 + 9t-2 - 9t-1 + 9 - 9t + 9t2 - 5t3 + t4

Conway Polynomial:

1 - 2z2 - z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{57, 4}

Jones Polynomial:

q-2 - 2q-1 + 4 - 5q + 7q2 - 8q3 + 8q4 - 8q5 + 6q6 - 4q7 + 3q8 - q9

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

{...}

A2 (sl(3)) Invariant:

q-6 + q-4 + 1 + q4 - q8 - 3q12 + q14 + q18 + q20 + q24 - q26

HOMFLY-PT Polynomial:

- 3a-6z2 - 4a-6z4 - a-6z6 + 2a-4 + 10a-4z2 + 12a-4z4 + 6a-4z6 + a-4z8 - 4a-2 - 13a-2z2 - 10a-2z4 - 2a-2z6 + 3 + 4z2 + z4

Kauffman Polynomial:

a-11z3 - 2a-10z2 + 3a-10z4 + a-9z - 3a-9z3 + 4a-9z5 - 3a-8z2 - 2a-8z4 + 4a-8z6 + 3a-7z - 5a-7z3 - 4a-7z5 + 4a-7z7 - a-6z2 + 3a-6z4 - 9a-6z6 + 4a-6z8 + a-5z + a-5z3 + 2a-5z5 - 8a-5z7 + 3a-5z9 + 2a-4 - 13a-4z2 + 31a-4z4 - 22a-4z6 + 2a-4z8 + a-4z10 - a-3z - 7a-3z3 + 28a-3z5 - 23a-3z7 + 5a-3z9 + 4a-2 - 23a-2z2 + 35a-2z4 - 15a-2z6 - a-2z8 + a-2z10 - 9a-1z3 + 18a-1z5 - 11a-1z7 + 2a-1z9 + 3 - 10z2 + 12z4 - 6z6 + z8

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

{-2, -2}

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 11179. 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                   2 1   j = 13                 4 2     j = 11               4 2       j = 9             4 4         j = 7           4 4           j = 5         3 4             j = 3       3 5               j = 1     1 2                 j = -1   1 3                   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, 179]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 179]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[20, 11, 21, 12], X[2, 13, 3, 14], X[6, 16, 7, 15], > X[8, 18, 9, 17], X[22, 20, 1, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 179]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -8, 4, -9, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -6, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 179]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 20, 2, 6, 8, 22, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 179]][t]
Out[6]=	-4 5 9 9 2 3 4 9 + t - -- + -- - - - 9 t + 9 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 179]][z]
Out[7]=	2 4 6 8 1 - 2 z - z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 179]}
In[9]:=	{KnotDet[Knot[11, Alternating, 179]], KnotSignature[Knot[11, Alternating, 179]]}
Out[9]=	{57, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 179]][q]
Out[10]=	-2 2 2 3 4 5 6 7 8 9 4 + q - - - 5 q + 7 q - 8 q + 8 q - 8 q + 6 q - 4 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 179]}
In[12]:=	A2Invariant[Knot[11, Alternating, 179]][q]
Out[12]=	-6 -4 4 8 12 14 18 20 24 26 1 + q + q + q - q - 3 q + q + q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 179]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 4 2 3 z 10 z 13 z 4 4 z 12 z 10 z z 3 + -- - -- + 4 z - ---- + ----- - ----- + z - ---- + ----- - ----- - -- + 4 2 6 4 2 6 4 2 6 a a a a a a a a a 6 6 8 6 z 2 z z > ---- - ---- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 179]][a, z]
Out[14]=	2 2 2 2 2 2 4 z 3 z z z 2 2 z 3 z z 13 z 23 z 3 + -- + -- + -- + --- + -- - -- - 10 z - ---- - ---- - -- - ----- - ----- + 4 2 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 z 3 z 5 z z 7 z 9 z 4 3 z 2 z 3 z 31 z > --- - ---- - ---- + -- - ---- - ---- + 12 z + ---- - ---- + ---- + ----- + 11 9 7 5 3 a 10 8 6 4 a a a a a a a a a 4 5 5 5 5 5 6 6 6 35 z 4 z 4 z 2 z 28 z 18 z 6 4 z 9 z 22 z > ----- + ---- - ---- + ---- + ----- + ----- - 6 z + ---- - ---- - ----- - 2 9 7 5 3 a 8 6 4 a a a a a a a a 6 7 7 7 7 8 8 8 9 9 15 z 4 z 8 z 23 z 11 z 8 4 z 2 z z 3 z 5 z > ----- + ---- - ---- - ----- - ----- + z + ---- + ---- - -- + ---- + ---- + 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a 9 10 10 2 z z z > ---- + --- + --- a 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 179]], Vassiliev[3][Knot[11, Alternating, 179]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, Alternating, 179]][q, t]
Out[16]=	3 3 5 1 1 1 3 q 2 q 3 q 5 7 5 q + 3 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 4 q t + 4 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 > 4 q t + 4 q t + 4 q t + 4 q t + 2 q t + 4 q t + 2 q t + 15 5 15 6 17 6 19 7 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a179

K11a178 K11a180