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

K11a269 K11a271

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 12 20 18 16 22 4 2 8 14

Alexander Polynomial:

- 2t-3 + 12t-2 - 32t-1 + 45 - 32t + 12t2 - 2t3

Conway Polynomial:

1 - 2z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{137, 0}

Jones Polynomial:

- q-5 + 4q-4 - 8q-3 + 14q-2 - 19q-1 + 22 - 22q + 19q2 - 14q3 + 9q4 - 4q5 + q6

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

{K11a80, ...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 3q-10 + 3q-8 + q-6 - 3q-4 + 4q-2 - 3 + 2q2 - q4 - 2q6 + 4q8 - 4q10 + 2q12 + 2q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4 + a-4z2 + a-4z4 - a-2 - 3a-2z2 - 2a-2z4 - a-2z6 - z2 - z4 - z6 + a2 + 2a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 3a-5z3 - 9a-5z5 + 4a-5z7 + a-4 - 7a-4z2 + 18a-4z4 - 22a-4z6 + 8a-4z8 + 2a-3z - 8a-3z3 + 17a-3z5 - 20a-3z7 + 8a-3z9 + a-2 - 14a-2z2 + 39a-2z4 - 35a-2z6 + 7a-2z8 + 3a-2z10 + 2a-1z - 16a-1z3 + 40a-1z5 - 39a-1z7 + 15a-1z9 - 6z2 + 24z4 - 25z6 + 7z8 + 3z10 - az3 + 3az5 - 8az7 + 7az9 - a2 + 3a2z2 - a2z4 - 9a2z6 + 8a2z8 + 3a3z3 - 10a3z5 + 7a3z7 + 2a4z2 - 6a4z4 + 4a4z6 - a5z3 + a5z5

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

{-2, -1}

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 11270. 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 = 13                       1 j = 11                     3   j = 9                   6 1   j = 7                 8 3     j = 5               11 6       j = 3             11 8         j = 1           11 11           j = -1         9 12             j = -3       5 10               j = -5     3 9                 j = -7   1 5                   j = -9   3                     j = -11 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, 270]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 270]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[18, 10, 19, 9], X[16, 11, 17, 12], X[22, 13, 1, 14], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 270]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, 7, -11, 8, -6, 9, -5, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 270]]
Out[5]=	DTCode[6, 10, 12, 20, 18, 16, 22, 4, 2, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 270]][t]
Out[6]=	2 12 32 2 3 45 - -- + -- - -- - 32 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 270]][z]
Out[7]=	2 6 1 - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 270]}
In[9]:=	{KnotDet[Knot[11, Alternating, 270]], KnotSignature[Knot[11, Alternating, 270]]}
Out[9]=	{137, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 270]][q]
Out[10]=	-5 4 8 14 19 2 3 4 5 6 22 - q + -- - -- + -- - -- - 22 q + 19 q - 14 q + 9 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 80], Knot[11, Alternating, 270]}
In[12]:=	A2Invariant[Knot[11, Alternating, 270]][q]
Out[12]=	-16 -14 2 3 3 -6 3 4 2 4 6 8 -3 - q + q + --- - --- + -- + q - -- + -- + 2 q - q - 2 q + 4 q - 12 10 8 4 2 q q q q q 10 12 14 16 18 > 4 q + 2 q + 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 270]][a, z]
Out[13]=	2 2 4 4 -4 -2 2 2 z 3 z 2 2 4 2 4 z 2 z 2 4 a - a + a - z + -- - ---- + 2 a z - a z - z + -- - ---- + 2 a z - 4 2 4 2 a a a a 6 6 z > z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 270]][a, z]
Out[14]=	2 2 3 -4 -2 2 2 z 2 z 2 7 z 14 z 2 2 4 2 3 z a + a - a + --- + --- - 6 z - ---- - ----- + 3 a z + 2 a z + ---- - 3 a 4 2 5 a a a a 3 3 4 4 4 8 z 16 z 3 3 3 5 3 4 2 z 18 z 39 z > ---- - ----- - a z + 3 a z - a z + 24 z - ---- + ----- + ----- - 3 a 6 4 2 a a a a 5 5 5 2 4 4 4 9 z 17 z 40 z 5 3 5 5 5 > a z - 6 a z - ---- + ----- + ----- + 3 a z - 10 a z + a z - 5 3 a a a 6 6 6 7 7 7 6 z 22 z 35 z 2 6 4 6 4 z 20 z 39 z > 25 z + -- - ----- - ----- - 9 a z + 4 a z + ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 8 z 7 z 2 8 8 z 15 z 9 > 8 a z + 7 a z + 7 z + ---- + ---- + 8 a z + ---- + ----- + 7 a z + 4 2 3 a a a a 10 10 3 z > 3 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 270]], Vassiliev[3][Knot[11, Alternating, 270]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, Alternating, 270]][q, t]
Out[16]=	12 1 3 1 5 3 9 5 10 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 9 3 3 2 5 2 5 3 7 3 7 4 > --- + 11 q t + 11 q t + 8 q t + 11 q t + 6 q t + 8 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a270

K11a269 K11a271