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

K11a270 K11a272

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 12 22 16 18 8 20 4 2 14

Alexander Polynomial:

3t-3 - 17t-2 + 40t-1 - 51 + 40t - 17t2 + 3t3

Conway Polynomial:

1 - z2 + z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{171, 2}

Jones Polynomial:

- q-2 + 5q-1 - 11 + 19q - 24q2 + 28q3 - 28q4 + 23q5 - 17q6 + 10q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 3q-4 - q-2 - 2 + 6q2 - 4q4 + 4q6 + q8 - 3q10 + 3q12 - 6q14 + 4q16 - q18 - 3q20 + 4q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 2a-6 - 5a-6z2 - 3a-6z4 + a-4 + 5a-4z2 + 5a-4z4 + 2a-4z6 - 2a-2z2 + a-2z6 + 1 - z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - a-9z + 4a-9z3 - 8a-9z5 + 4a-9z7 + a-8 - 6a-8z2 + 12a-8z4 - 17a-8z6 + 8a-8z8 + a-7z - a-7z3 + 7a-7z5 - 16a-7z7 + 9a-7z9 + 2a-6 - 17a-6z2 + 44a-6z4 - 44a-6z6 + 11a-6z8 + 4a-6z10 + 4a-5z - 9a-5z3 + 30a-5z5 - 44a-5z7 + 21a-5z9 + a-4 - 15a-4z2 + 46a-4z4 - 54a-4z6 + 18a-4z8 + 4a-4z10 + 2a-3z - 2a-3z3 + a-3z5 - 13a-3z7 + 12a-3z9 - 5a-2z2 + 12a-2z4 - 23a-2z6 + 15a-2z8 + 2a-1z3 - 13a-1z5 + 11a-1z7 + 1 - 4z4 + 5z6 + az5

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

{-1, -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=2 is the signature of 11271. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   7 1   j = 13                 10 3     j = 11               13 7       j = 9             15 10         j = 7           13 13           j = 5         11 15             j = 3       8 13               j = 1     4 12                 j = -1   1 7                   j = -3   4                     j = -5 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, 271]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 271]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[22, 8, 1, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[4, 18, 5, 17], X[2, 19, 3, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 271]]
Out[4]=	GaussCode[1, -10, 2, -9, 3, -1, 4, -7, 5, -2, 6, -3, 7, -11, 8, -5, 9, -6, 10, > -8, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 271]]
Out[5]=	DTCode[6, 10, 12, 22, 16, 18, 8, 20, 4, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 271]][t]
Out[6]=	3 17 40 2 3 -51 + -- - -- + -- + 40 t - 17 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 271]][z]
Out[7]=	2 4 6 1 - z + z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 271]}
In[9]:=	{KnotDet[Knot[11, Alternating, 271]], KnotSignature[Knot[11, Alternating, 271]]}
Out[9]=	{171, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 271]][q]
Out[10]=	-2 5 2 3 4 5 6 7 8 9 -11 - q + - + 19 q - 24 q + 28 q - 28 q + 23 q - 17 q + 10 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 271]}
In[12]:=	A2Invariant[Knot[11, Alternating, 271]][q]
Out[12]=	-6 3 -2 2 4 6 8 10 12 14 16 -2 - q + -- - q + 6 q - 4 q + 4 q + q - 3 q + 3 q - 6 q + 4 q - 4 q 18 20 22 24 28 > q - 3 q + 4 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 271]][a, z]
Out[13]=	2 2 2 2 4 4 6 6 -8 2 -4 z 5 z 5 z 2 z 4 3 z 5 z 2 z z 1 + a - -- + a + -- - ---- + ---- - ---- - z - ---- + ---- + ---- + -- 6 8 6 4 2 6 4 4 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 271]][a, z]
Out[14]=	2 2 2 2 2 -8 2 -4 z z 4 z 2 z z 6 z 17 z 15 z 5 z 1 + a + -- + a - -- + -- + --- + --- + --- - ---- - ----- - ----- - ---- + 6 9 7 5 3 10 8 6 4 2 a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 z z 9 z 2 z 2 z 4 2 z 12 z 44 z 46 z > ---- - -- - ---- - ---- + ---- - 4 z - ---- + ----- + ----- + ----- + 9 7 5 3 a 10 8 6 4 a a a a a a a a 4 5 5 5 5 5 6 6 12 z 8 z 7 z 30 z z 13 z 5 6 z 17 z > ----- - ---- + ---- + ----- + -- - ----- + a z + 5 z + --- - ----- - 2 9 7 5 3 a 10 8 a a a a a a a 6 6 6 7 7 7 7 7 8 44 z 54 z 23 z 4 z 16 z 44 z 13 z 11 z 8 z > ----- - ----- - ----- + ---- - ----- - ----- - ----- + ----- + ---- + 6 4 2 9 7 5 3 a 8 a a a a a a a a 8 8 8 9 9 9 10 10 11 z 18 z 15 z 9 z 21 z 12 z 4 z 4 z > ----- + ----- + ----- + ---- + ----- + ----- + ----- + ----- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 271]], Vassiliev[3][Knot[11, Alternating, 271]]}
Out[15]=	{-1, -3}
In[16]:=	Kh[Knot[11, Alternating, 271]][q, t]
Out[16]=	3 1 4 1 7 4 q 3 5 5 2 12 q + 8 q + ----- + ----- + ---- + --- + --- + 13 q t + 11 q t + 15 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 13 q t + 13 q t + 15 q t + 10 q t + 13 q t + 7 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 10 q t + 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a271

K11a270 K11a272