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

K11a166 K11a168

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 16 12 2 8 20 22 14 6

Alexander Polynomial:

- 2t-3 + 11t-2 - 26t-1 + 35 - 26t + 11t2 - 2t3

Conway Polynomial:

1 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a11, ...}

Determinant and Signature:

{113, 0}

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 11q-2 - 15q-1 + 18 - 18q + 16q2 - 12q3 + 8q4 - 4q5 + q6

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

{K11a11, ...}

A2 (sl(3)) Invariant:

- q-16 + q-12 - 2q-10 + 3q-8 + 2q-6 - 2q-4 + 3q-2 - 3 + q2 - q6 + 4q8 - 3q10 + q12 + q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 + a-2 - a-2z2 - 2a-2z4 - a-2z6 - 2 - 3z2 - 2z4 - z6 + 3a2 + 4a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 4a-5z3 - 10a-5z5 + 4a-5z7 - 4a-4z2 + 15a-4z4 - 20a-4z6 + 7a-4z8 + a-3z - 2a-3z3 + 8a-3z5 - 14a-3z7 + 6a-3z9 - a-2 - 6a-2z2 + 27a-2z4 - 27a-2z6 + 6a-2z8 + 2a-2z10 + 2a-1z - 12a-1z3 + 28a-1z5 - 26a-1z7 + 10a-1z9 - 2 + 3z2 + 6z4 - 9z6 + 3z8 + 2z10 + 2az - 6az3 + 5az5 - 4az7 + 4az9 - 3a2 + 9a2z2 - 10a2z4 + 4a2z8 + 2a3z - 2a3z3 - 4a3z5 + 4a3z7 - a4 + 4a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{0, -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=0 is the signature of 11167. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   5 1   j = 7                 7 3     j = 5               9 5       j = 3             9 7         j = 1           9 9           j = -1         7 10             j = -3       4 8               j = -5     2 7                 j = -7   1 4                   j = -9   2                     j = -11 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, 167]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 167]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[16, 8, 17, 7], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[22, 17, 1, 18], X[14, 20, 15, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 167]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -7, 5, -2, 6, -5, 7, -10, 8, -4, 9, -3, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 167]]
Out[5]=	DTCode[4, 10, 18, 16, 12, 2, 8, 20, 22, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 167]][t]
Out[6]=	2 11 26 2 3 35 - -- + -- - -- - 26 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 167]][z]
Out[7]=	4 6 1 - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 11], Knot[11, Alternating, 167]}
In[9]:=	{KnotDet[Knot[11, Alternating, 167]], KnotSignature[Knot[11, Alternating, 167]]}
Out[9]=	{113, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 167]][q]
Out[10]=	-5 3 6 11 15 2 3 4 5 6 18 - q + -- - -- + -- - -- - 18 q + 16 q - 12 q + 8 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 11], Knot[11, Alternating, 167]}
In[12]:=	A2Invariant[Knot[11, Alternating, 167]][q]
Out[12]=	-16 -12 2 3 2 2 3 2 6 8 10 12 -3 - q + q - --- + -- + -- - -- + -- + q - q + 4 q - 3 q + q + 10 8 6 4 2 q q q q q 14 16 18 > q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 167]][a, z]
Out[13]=	2 2 4 4 -2 2 4 2 z z 2 2 4 2 4 z 2 z -2 + a + 3 a - a - 3 z + -- - -- + 4 a z - a z - 2 z + -- - ---- + 4 2 4 2 a a a a 6 2 4 6 z > 2 a z - z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 167]][a, z]
Out[14]=	2 2 -2 2 4 z 2 z 3 5 2 4 z 6 z -2 - a - 3 a - a + -- + --- + 2 a z + 2 a z + a z + 3 z - ---- - ---- + 3 a 4 2 a a a 3 3 3 2 2 4 2 4 z 2 z 12 z 3 3 3 5 3 > 9 a z + 4 a z + ---- - ---- - ----- - 6 a z - 2 a z - 2 a z + 5 3 a a a 4 4 4 5 5 5 4 2 z 15 z 27 z 2 4 4 4 10 z 8 z 28 z > 6 z - ---- + ----- + ----- - 10 a z - 6 a z - ----- + ---- + ----- + 6 4 2 5 3 a a a a a a 6 6 6 7 5 3 5 5 5 6 z 20 z 27 z 4 6 4 z > 5 a z - 4 a z + a z - 9 z + -- - ----- - ----- + 3 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 9 14 z 26 z 7 3 7 8 7 z 6 z 2 8 6 z > ----- - ----- - 4 a z + 4 a z + 3 z + ---- + ---- + 4 a z + ---- + 3 a 4 2 3 a a a a 9 10 10 z 9 10 2 z > ----- + 4 a z + 2 z + ----- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 167]], Vassiliev[3][Knot[11, Alternating, 167]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 167]][q, t]
Out[16]=	10 1 2 1 4 2 7 4 8 -- + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a167

K11a166 K11a168