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

K11a209 K11a211

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 20 18 14 2 10 22 8 6

Alexander Polynomial:

4t-2 - 18t-1 + 29 - 18t + 4t2

Conway Polynomial:

1 - 2z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{73, 0}

Jones Polynomial:

q-6 - 2q-5 + 4q-4 - 7q-3 + 9q-2 - 11q-1 + 12 - 10q + 8q2 - 5q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-4z2 + a-2z4 + 2 + 2z2 + 2z4 - a2 - a2z2 + a2z4 - a4 - 2a4z2 + a6

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 2a-4z2 - 7a-4z4 + 3a-4z6 + 4a-3z3 - 9a-3z5 + 4a-3z7 - 4a-2z2 + 10a-2z4 - 10a-2z6 + 4a-2z8 + 2a-1z - 5a-1z3 + 11a-1z5 - 8a-1z7 + 3a-1z9 + 2 - 14z2 + 27z4 - 14z6 + 2z8 + z10 + 4az - 20az3 + 32az5 - 19az7 + 5az9 + a2 - 7a2z2 + 11a2z4 - 6a2z6 + a2z10 - 3a3z3 + 4a3z5 - 5a3z7 + 2a3z9 - a4 + 5a4z2 - 3a4z4 - 4a4z6 + 2a4z8 - 2a5z + 6a5z3 - 7a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

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=0 is the signature of 11210. 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                 5 2     j = 3               5 3       j = 1             7 5         j = -1           5 6           j = -3         4 6             j = -5       3 5               j = -7     1 4                 j = -9   1 3                   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, 210]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 210]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[20, 7, 21, 8], > X[18, 9, 19, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[10, 16, 11, 15], > X[22, 18, 1, 17], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 210]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -6, 8, -3, 9, -5, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 210]]
Out[5]=	DTCode[4, 12, 16, 20, 18, 14, 2, 10, 22, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 210]][t]
Out[6]=	4 18 2 29 + -- - -- - 18 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 210]][z]
Out[7]=	2 4 1 - 2 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 210]}
In[9]:=	{KnotDet[Knot[11, Alternating, 210]], KnotSignature[Knot[11, Alternating, 210]]}
Out[9]=	{73, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 210]][q]
Out[10]=	-6 2 4 7 9 11 2 3 4 5 12 + q - -- + -- - -- + -- - -- - 10 q + 8 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, 210]}
In[12]:=	A2Invariant[Knot[11, Alternating, 210]][q]
Out[12]=	-20 -18 -16 -14 3 -8 -6 -4 2 2 4 8 q + q - q + q - --- + q - q - q + -- + 3 q - q + 2 q - 10 2 q q 10 12 14 16 > 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 210]][a, z]
Out[13]=	2 4 2 4 6 2 z 2 2 4 2 4 z 2 4 2 - a - a + a + 2 z - -- - a z - 2 a z + 2 z + -- + a z 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 210]][a, z]
Out[14]=	2 2 2 4 6 2 z 5 2 2 z 4 z 2 2 2 + a - a - a + --- + 4 a z - 2 a z - 14 z + ---- - ---- - 7 a z + a 4 2 a a 3 3 3 4 2 6 2 2 z 4 z 5 z 3 3 3 5 3 > 5 a z + 4 a z - ---- + ---- - ---- - 20 a z - 3 a z + 6 a z + 5 3 a a a 4 4 5 5 5 4 7 z 10 z 2 4 4 4 6 4 z 9 z 11 z > 27 z - ---- + ----- + 11 a z - 3 a z - 4 a z + -- - ---- + ----- + 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 10 z 2 6 4 6 > 32 a z + 4 a z - 7 a z - 14 z + ---- - ----- - 6 a z - 4 a z + 4 2 a a 7 7 8 6 6 4 z 8 z 7 3 7 5 7 8 4 z 4 8 > a z + ---- - ---- - 19 a z - 5 a z + 2 a z + 2 z + ---- + 2 a z + 3 a 2 a a 9 3 z 9 3 9 10 2 10 > ---- + 5 a z + 2 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 210]], Vassiliev[3][Knot[11, Alternating, 210]]}
Out[15]=	{-2, 3}
In[16]:=	Kh[Knot[11, Alternating, 210]][q, t]
Out[16]=	6 1 1 1 3 1 4 3 5 - + 7 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 6 5 3 3 2 5 2 5 3 > ----- + ---- + --- + 5 q t + 5 q t + 3 q t + 5 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 K11a210

K11a209 K11a211