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

K11a284 K11a286

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 14 20 18 2 22 12 4 8

Alexander Polynomial:

- 2t-3 + 14t-2 - 38t-1 + 53 - 38t + 14t2 - 2t3

Conway Polynomial:

1 + 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a138, ...}

Determinant and Signature:

{161, 0}

Jones Polynomial:

q-6 - 5q-5 + 11q-4 - 17q-3 + 23q-2 - 26q-1 + 26 - 22q + 16q2 - 9q3 + 4q4 - q5

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

{K11a138, ...}

A2 (sl(3)) Invariant:

q-18 - 3q-16 + 2q-14 + 2q-12 - 5q-10 + 5q-8 - 2q-6 + 3q-2 - 3 + 5q2 - 4q4 + q6 + 3q8 - 4q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z2 + 2a-2z4 + 1 + z2 - z6 - a2z2 - a2z4 - a2z6 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 + 4a-3z3 - 11a-3z5 + 8a-3z7 - 4a-2z2 + 13a-2z4 - 18a-2z6 + 11a-2z8 + a-1z - 4a-1z3 + 11a-1z5 - 15a-1z7 + 10a-1z9 + 1 - 12z2 + 38z4 - 39z6 + 11z8 + 4z10 + 3az - 17az3 + 41az5 - 47az7 + 20az9 - 8a2z2 + 32a2z4 - 40a2z6 + 10a2z8 + 4a2z10 + 3a3z - 6a3z3 + 9a3z5 - 19a3z7 + 10a3z9 - a4z2 + 11a4z4 - 22a4z6 + 10a4z8 + a5z + 2a5z3 - 9a5z5 + 5a5z7 - a6z4 + a6z6

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

{0, 0}

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 11285. 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                     3   j = 7                   6 1   j = 5                 10 3     j = 3               12 6       j = 1             14 10         j = -1           13 13           j = -3         10 13             j = -5       7 13               j = -7     4 10                 j = -9   1 7                   j = -11   4                     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, 285]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 285]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[20, 9, 21, 10], X[18, 12, 19, 11], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[12, 18, 13, 17], X[4, 19, 5, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 285]]
Out[4]=	GaussCode[1, -7, 2, -10, 3, -1, 4, -11, 5, -2, 6, -9, 7, -4, 8, -3, 9, -6, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 285]]
Out[5]=	DTCode[6, 10, 16, 14, 20, 18, 2, 22, 12, 4, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 285]][t]
Out[6]=	2 14 38 2 3 53 - -- + -- - -- - 38 t + 14 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 285]][z]
Out[7]=	4 6 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 138], Knot[11, Alternating, 285]}
In[9]:=	{KnotDet[Knot[11, Alternating, 285]], KnotSignature[Knot[11, Alternating, 285]]}
Out[9]=	{161, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 285]][q]
Out[10]=	-6 5 11 17 23 26 2 3 4 5 26 + q - -- + -- - -- + -- - -- - 22 q + 16 q - 9 q + 4 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, 138], Knot[11, Alternating, 285]}
In[12]:=	A2Invariant[Knot[11, Alternating, 285]][q]
Out[12]=	-18 3 2 2 5 5 2 3 2 4 6 8 -3 + q - --- + --- + --- - --- + -- - -- + -- + 5 q - 4 q + q + 3 q - 16 14 12 10 8 6 2 q q q q q q q 10 12 14 16 > 4 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 285]][a, z]
Out[13]=	2 2 4 2 z z 2 2 2 z 2 4 4 4 6 2 6 1 + z - -- + -- - a z + ---- - a z + a z - z - a z 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 285]][a, z]
Out[14]=	2 2 3 z 3 5 2 z 4 z 2 2 4 2 z 1 + - + 3 a z + 3 a z + a z - 12 z + -- - ---- - 8 a z - a z - -- + a 4 2 5 a a a 3 3 4 4 4 z 4 z 3 3 3 5 3 4 5 z 13 z > ---- - ---- - 17 a z - 6 a z + 2 a z + 38 z - ---- + ----- + 3 a 4 2 a a a 5 5 5 2 4 4 4 6 4 z 11 z 11 z 5 3 5 > 32 a z + 11 a z - a z + -- - ----- + ----- + 41 a z + 9 a z - 5 3 a a a 6 6 7 5 5 6 4 z 18 z 2 6 4 6 6 6 8 z > 9 a z - 39 z + ---- - ----- - 40 a z - 22 a z + a z + ---- - 4 2 3 a a a 7 8 15 z 7 3 7 5 7 8 11 z 2 8 > ----- - 47 a z - 19 a z + 5 a z + 11 z + ----- + 10 a z + a 2 a 9 4 8 10 z 9 3 9 10 2 10 > 10 a z + ----- + 20 a z + 10 a z + 4 z + 4 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 285]], Vassiliev[3][Knot[11, Alternating, 285]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 285]][q, t]
Out[16]=	13 1 4 1 7 4 10 7 13 -- + 14 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 10 13 13 3 3 2 5 2 5 3 > ----- + ---- + --- + 10 q t + 12 q t + 6 q t + 10 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a285

K11a284 K11a286