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

K11a170 K11a172

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 20 2 22 6 8 12 14 16

Alexander Polynomial:

- t-4 + 7t-3 - 21t-2 + 39t-1 - 47 + 39t - 21t2 + 7t3 - t4

Conway Polynomial:

1 + 2z2 + z4 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{183, 2}

Jones Polynomial:

q-3 - 5q-2 + 11q-1 - 18 + 26q - 29q2 + 30q3 - 26q4 + 19q5 - 12q6 + 5q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-8 - 3q-6 + 3q-4 - 3q-2 - 1 + 6q2 - 4q4 + 8q6 - 2q8 + q10 + q12 - 6q14 + 4q16 - 3q18 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 - 2a-4 + 2a-4z2 + 5a-4z4 + 2a-4z6 + 4a-2 + a-2z2 - 5a-2z4 - 4a-2z6 - a-2z8 - 1 + 2z4 + z6

Kauffman Polynomial:

a-9z5 - 3a-8z4 + 5a-8z6 - a-7z + 6a-7z3 - 15a-7z5 + 12a-7z7 - 4a-6z2 + 12a-6z4 - 23a-6z6 + 16a-6z8 - a-5z + 17a-5z3 - 28a-5z5 + a-5z7 + 11a-5z9 - 2a-4 - 8a-4z2 + 47a-4z4 - 71a-4z6 + 29a-4z8 + 3a-4z10 - a-3z + 21a-3z3 - 18a-3z5 - 23a-3z7 + 20a-3z9 - 4a-2 - 6a-2z2 + 47a-2z4 - 65a-2z6 + 23a-2z8 + 3a-2z10 - a-1z + 14a-1z3 - 15a-1z5 - 7a-1z7 + 9a-1z9 - 1 - 2z2 + 14z4 - 21z6 + 10z8 + 4az3 - 9az5 + 5az7 - a2z4 + a2z6

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=2 is the signature of 11171. 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 = 17                       1 j = 15                     4   j = 13                   8 1   j = 11                 11 4     j = 9               15 8       j = 7             15 11         j = 5           14 15           j = 3         12 15             j = 1       7 15               j = -1     4 11                 j = -3   1 7                   j = -5   4                     j = -7 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, 171]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 171]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[6, 14, 7, 13], X[8, 15, 9, 16], > X[12, 18, 13, 17], X[14, 19, 15, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 171]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -8, 5, -2, 6, -9, 7, -10, 8, -11, 9, -3, 10, > -4, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 171]]
Out[5]=	DTCode[4, 10, 18, 20, 2, 22, 6, 8, 12, 14, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 171]][t]
Out[6]=	-4 7 21 39 2 3 4 -47 - t + -- - -- + -- + 39 t - 21 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 171]][z]
Out[7]=	2 4 6 8 1 + 2 z + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 171]}
In[9]:=	{KnotDet[Knot[11, Alternating, 171]], KnotSignature[Knot[11, Alternating, 171]]}
Out[9]=	{183, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 171]][q]
Out[10]=	-3 5 11 2 3 4 5 6 7 8 -18 + q - -- + -- + 26 q - 29 q + 30 q - 26 q + 19 q - 12 q + 5 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 171]}
In[12]:=	A2Invariant[Knot[11, Alternating, 171]][q]
Out[12]=	-8 3 3 3 2 4 6 8 10 12 14 -1 + q - -- + -- - -- + 6 q - 4 q + 8 q - 2 q + q + q - 6 q + 6 4 2 q q q 16 18 22 24 > 4 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 171]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 8 2 4 z 2 z z 4 z 5 z 5 z 6 2 z 4 z z -1 - -- + -- - -- + ---- + -- + 2 z - -- + ---- - ---- + z + ---- - ---- - -- 4 2 6 4 2 6 4 2 4 2 2 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 171]][a, z]
Out[14]=	2 2 2 3 3 2 4 z z z z 2 4 z 8 z 6 z 6 z 17 z -1 - -- - -- - -- - -- - -- - - - 2 z - ---- - ---- - ---- + ---- + ----- + 4 2 7 5 3 a 6 4 2 7 5 a a a a a a a a a a 3 3 4 4 4 4 21 z 14 z 3 4 3 z 12 z 47 z 47 z 2 4 > ----- + ----- + 4 a z + 14 z - ---- + ----- + ----- + ----- - a z + 3 a 8 6 4 2 a a a a a 5 5 5 5 5 6 6 z 15 z 28 z 18 z 15 z 5 6 5 z 23 z > -- - ----- - ----- - ----- - ----- - 9 a z - 21 z + ---- - ----- - 9 7 5 3 a 8 6 a a a a a a 6 6 7 7 7 7 71 z 65 z 2 6 12 z z 23 z 7 z 7 8 > ----- - ----- + a z + ----- + -- - ----- - ---- + 5 a z + 10 z + 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 16 z 29 z 23 z 11 z 20 z 9 z 3 z 3 z > ----- + ----- + ----- + ----- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 171]], Vassiliev[3][Knot[11, Alternating, 171]]}
Out[15]=	{2, 2}
In[16]:=	Kh[Knot[11, Alternating, 171]][q, t]
Out[16]=	3 1 4 1 7 4 11 7 q 3 15 q + 12 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 15 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 14 q t + 15 q t + 15 q t + 11 q t + 15 q t + 8 q t + 11 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a171

K11a170 K11a172