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

K11a277 K11a279

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 12 4 18 22 20 2 8 14

Alexander Polynomial:

- 2t-3 + 12t-2 - 33t-1 + 47 - 33t + 12t2 - 2t3

Conway Polynomial:

1 - 3z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{141, 0}

Jones Polynomial:

q-4 - 4q-3 + 10q-2 - 15q-1 + 20 - 23q + 22q2 - 19q3 + 14q4 - 8q5 + 4q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + 3q-8 + 3q-6 - 4q-4 + 4q-2 - 3 - q2 + q4 - 3q6 + 4q8 - 3q10 + q12 + 3q14 - 3q16 + 2q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4 + 2a-4z2 + 2a-4z4 - a-2z2 - a-2z4 - a-2z6 - 2 - 4z2 - 2z4 - z6 + 2a2 + a2z2 + a2z4

Kauffman Polynomial:

2a-7z3 - 3a-7z5 + a-7z7 - 5a-6z2 + 15a-6z4 - 14a-6z6 + 4a-6z8 + a-5z - 4a-5z3 + 17a-5z5 - 19a-5z7 + 6a-5z9 + a-4 - 17a-4z2 + 46a-4z4 - 37a-4z6 + 4a-4z8 + 3a-4z10 + 5a-3z - 14a-3z3 + 35a-3z5 - 44a-3z7 + 16a-3z9 - 13a-2z2 + 44a-2z4 - 54a-2z6 + 15a-2z8 + 3a-2z10 + 4a-1z - 9a-1z5 - 9a-1z7 + 10a-1z9 - 2 + 4z2 + 3z4 - 21z6 + 15z8 + 8az3 - 20az5 + 15az7 - 2a2 + 5a2z2 - 9a2z4 + 10a2z6 + 4a3z5 + a4z4

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

{-3, -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 11278. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 9 3     j = 7               10 5       j = 5             12 9         j = 3           11 10           j = 1         9 12             j = -1       7 12               j = -3     3 8                 j = -5   1 7                   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, 278]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 278]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[4, 10, 5, 9], X[18, 11, 19, 12], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[2, 17, 3, 18], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 278]]
Out[4]=	GaussCode[1, -9, 2, -5, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 278]]
Out[5]=	DTCode[6, 10, 16, 12, 4, 18, 22, 20, 2, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 278]][t]
Out[6]=	2 12 33 2 3 47 - -- + -- - -- - 33 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 278]][z]
Out[7]=	2 6 1 - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 278]}
In[9]:=	{KnotDet[Knot[11, Alternating, 278]], KnotSignature[Knot[11, Alternating, 278]]}
Out[9]=	{141, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 278]][q]
Out[10]=	-4 4 10 15 2 3 4 5 6 7 20 + q - -- + -- - -- - 23 q + 22 q - 19 q + 14 q - 8 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, 278]}
In[12]:=	A2Invariant[Knot[11, Alternating, 278]][q]
Out[12]=	-12 2 3 3 4 4 2 4 6 8 10 12 -3 + q - --- + -- + -- - -- + -- - q + q - 3 q + 4 q - 3 q + q + 10 8 6 4 2 q q q q q 14 16 18 20 22 > 3 q - 3 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 278]][a, z]
Out[13]=	2 2 2 4 4 -4 2 2 z 2 z z 2 2 4 2 z z 2 4 -2 + a + 2 a - 4 z - -- + ---- - -- + a z - 2 z + ---- - -- + a z - 6 4 2 4 2 a a a a a 6 6 z > z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 278]][a, z]
Out[14]=	2 2 2 -4 2 z 5 z 4 z 2 5 z 17 z 13 z 2 2 -2 + a - 2 a + -- + --- + --- + 4 z - ---- - ----- - ----- + 5 a z + 5 3 a 6 4 2 a a a a a 3 3 3 4 4 4 2 z 4 z 14 z 3 4 15 z 46 z 44 z 2 4 > ---- - ---- - ----- + 8 a z + 3 z + ----- + ----- + ----- - 9 a z + 7 5 3 6 4 2 a a a a a a 5 5 5 5 6 4 4 3 z 17 z 35 z 9 z 5 3 5 6 14 z > a z - ---- + ----- + ----- - ---- - 20 a z + 4 a z - 21 z - ----- - 7 5 3 a 6 a a a a 6 6 7 7 7 7 37 z 54 z 2 6 z 19 z 44 z 9 z 7 8 > ----- - ----- + 10 a z + -- - ----- - ----- - ---- + 15 a z + 15 z + 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 4 z 4 z 15 z 6 z 16 z 10 z 3 z 3 z > ---- + ---- + ----- + ---- + ----- + ----- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 278]], Vassiliev[3][Knot[11, Alternating, 278]]}
Out[15]=	{-3, -1}
In[16]:=	Kh[Knot[11, Alternating, 278]][q, t]
Out[16]=	12 1 3 1 7 3 8 7 -- + 9 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 12 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 11 q t + 10 q t + 12 q t + 9 q t + 10 q t + 5 q t + 9 q t + 9 5 11 5 11 6 13 6 15 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a278

K11a277 K11a279