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

K11a200 K11a202

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 14 20 18 2 6 22 10 8

Alexander Polynomial:

4t-2 - 20t-1 + 33 - 20t + 4t2

Conway Polynomial:

1 - 4z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a103, ...}

Determinant and Signature:

{81, 0}

Jones Polynomial:

q-6 - 2q-5 + 5q-4 - 8q-3 + 10q-2 - 13q-1 + 13 - 11q + 9q2 - 5q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 - q-16 + 2q-14 + q-12 - 3q-10 + q-8 - 2q-6 - 2q-4 + q-2 - 1 + 3q2 - q4 + q6 + 3q8 - 2q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2 + a-2z4 + 1 + z2 + 2z4 - 2a2 - 2a2z2 + a2z4 - 2a4z2 + a6

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 + 3a-3z3 - 8a-3z5 + 4a-3z7 - a-2 - a-2z2 + 6a-2z4 - 8a-2z6 + 4a-2z8 + 4a-1z - 6a-1z3 + 8a-1z5 - 6a-1z7 + 3a-1z9 + 1 - 18z2 + 35z4 - 20z6 + 4z8 + z10 + 12az - 32az3 + 39az5 - 22az7 + 6az9 + 2a2 - 21a2z2 + 33a2z4 - 19a2z6 + 3a2z8 + a2z10 + 8a3z - 18a3z3 + 16a3z5 - 10a3z7 + 3a3z9 - 3a4z2 + 7a4z4 - 9a4z6 + 3a4z8 + 3a5z3 - 6a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

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

{-4, 4}

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 11201. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   3 1   j = 5                 6 2     j = 3               5 3       j = 1             8 6         j = -1           6 6           j = -3         4 7             j = -5       4 6               j = -7     1 4                 j = -9   1 4                   j = -11   1                     j = -13 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, 201]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 201]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[14, 8, 15, 7], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[2, 13, 3, 14], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 201]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -6, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 201]]
Out[5]=	DTCode[4, 12, 16, 14, 20, 18, 2, 6, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 201]][t]
Out[6]=	4 20 2 33 + -- - -- - 20 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 201]][z]
Out[7]=	2 4 1 - 4 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 103], Knot[11, Alternating, 201]}
In[9]:=	{KnotDet[Knot[11, Alternating, 201]], KnotSignature[Knot[11, Alternating, 201]]}
Out[9]=	{81, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 201]][q]
Out[10]=	-6 2 5 8 10 13 2 3 4 5 13 + q - -- + -- - -- + -- - -- - 11 q + 9 q - 5 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 201]}
In[12]:=	A2Invariant[Knot[11, Alternating, 201]][q]
Out[12]=	-20 -18 -16 2 -12 3 -8 2 2 -2 2 4 -1 + q + q - q + --- + q - --- + q - -- - -- + q + 3 q - q + 14 10 6 4 q q q q 6 8 10 12 14 16 > q + 3 q - 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 201]][a, z]
Out[13]=	2 4 -2 2 6 2 z 2 2 4 2 4 z 2 4 1 + a - 2 a + a + z - -- - 2 a z - 2 a z + 2 z + -- + a z 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 201]][a, z]
Out[14]=	2 2 -2 2 6 4 z 3 2 3 z z 2 2 1 - a + 2 a - a + --- + 12 a z + 8 a z - 18 z + ---- - -- - 21 a z - a 4 2 a a 3 3 3 4 2 6 2 2 z 3 z 6 z 3 3 3 5 3 > 3 a z + 4 a z - ---- + ---- - ---- - 32 a z - 18 a z + 3 a z + 5 3 a a a 4 4 5 5 5 4 7 z 6 z 2 4 4 4 6 4 z 8 z 8 z > 35 z - ---- + ---- + 33 a z + 7 a z - 4 a z + -- - ---- + ---- + 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 8 z 2 6 4 6 > 39 a z + 16 a z - 6 a z - 20 z + ---- - ---- - 19 a z - 9 a z + 4 2 a a 7 7 8 6 6 4 z 6 z 7 3 7 5 7 8 4 z > a z + ---- - ---- - 22 a z - 10 a z + 2 a z + 4 z + ---- + 3 a 2 a a 9 2 8 4 8 3 z 9 3 9 10 2 10 > 3 a z + 3 a z + ---- + 6 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 201]], Vassiliev[3][Knot[11, Alternating, 201]]}
Out[15]=	{-4, 4}
In[16]:=	Kh[Knot[11, Alternating, 201]][q, t]
Out[16]=	6 1 1 1 4 1 4 4 6 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 4 7 6 3 3 2 5 2 5 3 > ----- + ---- + --- + 6 q t + 5 q t + 3 q t + 6 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a201

K11a200 K11a202