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

K11a279 K11a281

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X22,14,1,13 X20,16,21,15 X4,17,5,18 X8,19,9,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 12 18 2 22 20 4 8 14

Alexander Polynomial:

6t-2 - 26t-1 + 41 - 26t + 6t2

Conway Polynomial:

1 - 2z2 + 6z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{105, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 11q-3 + 14q-2 - 17q-1 + 17 - 14q + 11q2 - 6q3 + 3q4 - q5

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

{K11a16, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-4z2 - a-2z2 + a-2z4 + 2 + 3z2 + 3z4 - a2 + 2a2z4 - a4 - 3a4z2 + a6

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + a-4z2 - 6a-4z4 + 3a-4z6 + 4a-3z3 - 10a-3z5 + 5a-3z7 - 11a-2z2 + 25a-2z4 - 20a-2z6 + 7a-2z8 + 3a-1z - 10a-1z3 + 23a-1z5 - 17a-1z7 + 6a-1z9 + 2 - 24z2 + 50z4 - 30z6 + 5z8 + 2z10 + 7az - 26az3 + 44az5 - 32az7 + 10az9 + a2 - 11a2z2 + 19a2z4 - 15a2z6 + 2a2z8 + 2a2z10 + 3a3z - 5a3z3 + 2a3z5 - 7a3z7 + 4a3z9 - a4 + 4a4z2 - 3a4z4 - 7a4z6 + 4a4z8 - a5z + 5a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + 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 11280. 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                   4 1   j = 5                 7 2     j = 3               7 4       j = 1             10 7         j = -1           8 8           j = -3         6 9             j = -5       5 8               j = -7     2 6                 j = -9   1 5                   j = -11   2                     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, 280]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 280]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[4, 17, 5, 18], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 280]]
Out[4]=	GaussCode[1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 280]]
Out[5]=	DTCode[6, 10, 16, 12, 18, 2, 22, 20, 4, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 280]][t]
Out[6]=	6 26 2 41 + -- - -- - 26 t + 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 280]][z]
Out[7]=	2 4 1 - 2 z + 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 280]}
In[9]:=	{KnotDet[Knot[11, Alternating, 280]], KnotSignature[Knot[11, Alternating, 280]]}
Out[9]=	{105, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 280]][q]
Out[10]=	-6 3 7 11 14 17 2 3 4 5 17 + q - -- + -- - -- + -- - -- - 14 q + 11 q - 6 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, 16], Knot[11, Alternating, 280]}
In[12]:=	A2Invariant[Knot[11, Alternating, 280]][q]
Out[12]=	-20 -18 2 2 -12 4 2 2 -4 2 2 4 -1 + q + q - --- + --- + q - --- + -- - -- - q + -- + 4 q - 2 q + 16 14 10 8 6 2 q q q q q q 6 8 10 12 14 16 > q + 3 q - 3 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 280]][a, z]
Out[13]=	2 2 4 2 4 6 2 z z 4 2 4 z 2 4 2 - a - a + a + 3 z - -- - -- - 3 a z + 3 z + -- + 2 a z 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 280]][a, z]
Out[14]=	2 2 2 4 6 3 z 3 5 2 z 11 z 2 + a - a - a + --- + 7 a z + 3 a z - a z - 24 z + -- - ----- - a 4 2 a a 3 3 3 2 2 4 2 6 2 2 z 4 z 10 z 3 3 3 > 11 a z + 4 a z + 3 a z - ---- + ---- - ----- - 26 a z - 5 a z + 5 3 a a a 4 4 5 5 3 4 6 z 25 z 2 4 4 4 6 4 z > 5 a z + 50 z - ---- + ----- + 19 a z - 3 a z - 3 a z + -- - 4 2 5 a a a 5 5 6 6 10 z 23 z 5 3 5 5 5 6 3 z 20 z > ----- + ----- + 44 a z + 2 a z - 8 a z - 30 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 5 z 17 z 7 3 7 5 7 > 15 a z - 7 a z + a z + ---- - ----- - 32 a z - 7 a z + 3 a z + 3 a a 8 9 8 7 z 2 8 4 8 6 z 9 3 9 10 > 5 z + ---- + 2 a z + 4 a z + ---- + 10 a z + 4 a z + 2 z + 2 a a 2 10 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 280]], Vassiliev[3][Knot[11, Alternating, 280]]}
Out[15]=	{-2, 3}
In[16]:=	Kh[Knot[11, Alternating, 280]][q, t]
Out[16]=	8 1 2 1 5 2 6 5 8 - + 10 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 6 9 8 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 7 q t + 4 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a280

K11a279 K11a281