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

K11a161 K11a163

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 16 2 6 20 22 12 8

Alexander Polynomial:

- t-4 + 7t-3 - 20t-2 + 35t-1 - 41 + 35t - 20t2 + 7t3 - t4

Conway Polynomial:

1 + 2z2 + 2z4 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{167, -2}

Jones Polynomial:

- q-8 + 4q-7 - 10q-6 + 17q-5 - 23q-4 + 27q-3 - 27q-2 + 24q-1 - 17 + 11q - 5q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 + q-22 - 3q-18 + 4q-16 - 4q-14 + 2q-12 + 2q-10 - 3q-8 + 6q-6 - 5q-4 + 5q-2 - 2q2 + 3q4 - 3q6 + q8

HOMFLY-PT Polynomial:

2z4 + z6 + a2 - a2z2 - 5a2z4 - 4a2z6 - a2z8 + a4 + 5a4z2 + 6a4z4 + 2a4z6 - a6 - 2a6z2 - a6z4

Kauffman Polynomial:

- a-2z4 + a-2z6 + 3a-1z3 - 9a-1z5 + 5a-1z7 - z2 + 13z4 - 22z6 + 10z8 - az + 8az3 - 7az5 - 11az7 + 9az9 - a2 - 7a2z2 + 39a2z4 - 54a2z6 + 18a2z8 + 3a2z10 - 3a3z + 13a3z3 - 2a3z5 - 27a3z7 + 18a3z9 + a4 - 13a4z2 + 42a4z4 - 53a4z6 + 20a4z8 + 3a4z10 - 5a5z + 18a5z3 - 18a5z5 - 2a5z7 + 9a5z9 + a6 - 6a6z2 + 13a6z4 - 18a6z6 + 12a6z8 - 3a7z + 9a7z3 - 13a7z5 + 9a7z7 + a8z2 - 4a8z4 + 4a8z6 - a9z3 + a9z5

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

{2, -3}

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 11162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     4   j = 3                   7 1   j = 1                 10 4     j = -1               14 7       j = -3             14 11         j = -5           13 13           j = -7         10 14             j = -9       7 13               j = -11     3 10                 j = -13   1 7                   j = -15   3                     j = -17 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, 162]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 162]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 7, 15, 8], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[6, 13, 7, 14], X[20, 16, 21, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 162]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, -11, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 162]]
Out[5]=	DTCode[4, 10, 18, 14, 16, 2, 6, 20, 22, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 162]][t]
Out[6]=	-4 7 20 35 2 3 4 -41 - t + -- - -- + -- + 35 t - 20 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 162]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 162]}
In[9]:=	{KnotDet[Knot[11, Alternating, 162]], KnotSignature[Knot[11, Alternating, 162]]}
Out[9]=	{167, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 162]][q]
Out[10]=	-8 4 10 17 23 27 27 24 2 3 -17 - q + -- - -- + -- - -- + -- - -- + -- + 11 q - 5 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 162]}
In[12]:=	A2Invariant[Knot[11, Alternating, 162]][q]
Out[12]=	-24 -22 3 4 4 2 2 3 6 5 5 2 4 -q + q - --- + --- - --- + --- + --- - -- + -- - -- + -- - 2 q + 3 q - 18 16 14 12 10 8 6 4 2 q q q q q q q q q 6 8 > 3 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 162]][a, z]
Out[13]=	2 4 6 2 2 4 2 6 2 4 2 4 4 4 6 4 a + a - a - a z + 5 a z - 2 a z + 2 z - 5 a z + 6 a z - a z + 6 2 6 4 6 2 8 > z - 4 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 162]][a, z]
Out[14]=	2 4 6 3 5 7 2 2 2 4 2 -a + a + a - a z - 3 a z - 5 a z - 3 a z - z - 7 a z - 13 a z - 3 6 2 8 2 3 z 3 3 3 5 3 7 3 9 3 > 6 a z + a z + ---- + 8 a z + 13 a z + 18 a z + 9 a z - a z + a 4 5 4 z 2 4 4 4 6 4 8 4 9 z 5 > 13 z - -- + 39 a z + 42 a z + 13 a z - 4 a z - ---- - 7 a z - 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 2 a z - 18 a z - 13 a z + a z - 22 z + -- - 54 a z - 53 a z - 2 a 7 6 6 8 6 5 z 7 3 7 5 7 7 7 > 18 a z + 4 a z + ---- - 11 a z - 27 a z - 2 a z + 9 a z + a 8 2 8 4 8 6 8 9 3 9 5 9 > 10 z + 18 a z + 20 a z + 12 a z + 9 a z + 18 a z + 9 a z + 2 10 4 10 > 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 162]], Vassiliev[3][Knot[11, Alternating, 162]]}
Out[15]=	{2, -3}
In[16]:=	Kh[Knot[11, Alternating, 162]][q, t]
Out[16]=	11 14 1 3 1 7 3 10 7 13 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 10 14 13 13 14 7 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 10 q t + 4 q t + 7 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a162

K11a161 K11a163