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

K11a280 K11a282

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 12 20 2 18 4 22 8 14

Alexander Polynomial:

- t-4 + 6t-3 - 18t-2 + 33t-1 - 39 + 33t - 18t2 + 6t3 - t4

Conway Polynomial:

1 - z2 - 2z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a19, K11a25, ...}

Determinant and Signature:

{155, 2}

Jones Polynomial:

- q-4 + 4q-3 - 9q-2 + 16q-1 - 21 + 25q - 25q2 + 22q3 - 17q4 + 10q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 - 2q-6 + 5q-4 - 2q-2 + 3 + 2q2 - 4q4 + 4q6 - 6q8 + 3q10 - q12 - 2q14 + 4q16 - 2q18 + q20

HOMFLY-PT Polynomial:

2a-4 + 4a-4z2 + 3a-4z4 + a-4z6 - 5a-2 - 12a-2z2 - 11a-2z4 - 5a-2z6 - a-2z8 + 5 + 9z2 + 7z4 + 2z6 - a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 + 4a-7z5 + 3a-6z2 - 7a-6z4 + 10a-6z6 - 3a-5z + 14a-5z3 - 24a-5z5 + 17a-5z7 + 2a-4 - 7a-4z2 + 19a-4z4 - 33a-4z6 + 19a-4z8 - 6a-3z + 22a-3z3 - 25a-3z5 - 8a-3z7 + 12a-3z9 + 5a-2 - 28a-2z2 + 71a-2z4 - 81a-2z6 + 24a-2z8 + 3a-2z10 - 5a-1z + 12a-1z3 + 12a-1z5 - 41a-1z7 + 18a-1z9 + 5 - 25z2 + 59z4 - 51z6 + 9z8 + 3z10 - 3az + 7az3 + 6az5 - 15az7 + 6az9 + a2 - 7a2z2 + 15a2z4 - 13a2z6 + 4a2z8 - a3z + 3a3z3 - 3a3z5 + a3z7

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

{-1, -2}

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 11281. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   7 1   j = 9                 10 3     j = 7               12 7       j = 5             13 10         j = 3           12 12           j = 1         10 14             j = -1       6 11               j = -3     3 10                 j = -5   1 6                   j = -7   3                     j = -9 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, 281]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 281]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[12, 8, 13, 7], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[4, 16, 5, 15], > X[22, 17, 1, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 281]]
Out[4]=	GaussCode[1, -6, 2, -8, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 281]]
Out[5]=	DTCode[6, 10, 16, 12, 20, 2, 18, 4, 22, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 281]][t]
Out[6]=	-4 6 18 33 2 3 4 -39 - t + -- - -- + -- + 33 t - 18 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 281]][z]
Out[7]=	2 4 6 8 1 - z - 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 19], Knot[11, Alternating, 25], > Knot[11, Alternating, 281]}
In[9]:=	{KnotDet[Knot[11, Alternating, 281]], KnotSignature[Knot[11, Alternating, 281]]}
Out[9]=	{155, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 281]][q]
Out[10]=	-4 4 9 16 2 3 4 5 6 7 -21 - q + -- - -- + -- + 25 q - 25 q + 22 q - 17 q + 10 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 281]}
In[12]:=	A2Invariant[Knot[11, Alternating, 281]][q]
Out[12]=	-12 -10 2 5 2 2 4 6 8 10 12 3 - q + q - -- + -- - -- + 2 q - 4 q + 4 q - 6 q + 3 q - q - 6 4 2 q q q 14 16 18 20 > 2 q + 4 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 281]][a, z]
Out[13]=	2 2 4 4 2 5 2 2 4 z 12 z 2 2 4 3 z 11 z 5 + -- - -- - a + 9 z + ---- - ----- - 2 a z + 7 z + ---- - ----- - 4 2 4 2 4 2 a a a a a a 6 6 8 2 4 6 z 5 z z > a z + 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 281]][a, z]
Out[14]=	2 2 2 5 2 3 z 6 z 5 z 3 2 3 z 7 z 5 + -- + -- + a - --- - --- - --- - 3 a z - a z - 25 z + ---- - ---- - 4 2 5 3 a 6 4 a a a a a a 2 3 3 3 4 28 z 2 2 14 z 22 z 12 z 3 3 3 4 z > ----- - 7 a z + ----- + ----- + ----- + 7 a z + 3 a z + 59 z + -- - 2 5 3 a 8 a a a a 4 4 4 5 5 5 5 7 z 19 z 71 z 2 4 4 z 24 z 25 z 12 z 5 > ---- + ----- + ----- + 15 a z + ---- - ----- - ----- + ----- + 6 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 10 z 33 z 81 z 2 6 17 z 8 z 41 z > 3 a z - 51 z + ----- - ----- - ----- - 13 a z + ----- - ---- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 19 z 24 z 2 8 12 z 18 z 9 > 15 a z + a z + 9 z + ----- + ----- + 4 a z + ----- + ----- + 6 a z + 4 2 3 a a a a 10 10 3 z > 3 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 281]], Vassiliev[3][Knot[11, Alternating, 281]]}
Out[15]=	{-1, -2}
In[16]:=	Kh[Knot[11, Alternating, 281]][q, t]
Out[16]=	3 1 3 1 6 3 10 6 11 14 q + 12 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 10 q 3 5 5 2 7 2 7 3 9 3 > ---- + 12 q t + 13 q t + 10 q t + 12 q t + 7 q t + 10 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a281

K11a280 K11a282