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

K11a248 K11a250

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X22,11,1,12 X20,13,21,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 10 18 16 22 20 4 2 14 12

Alexander Polynomial:

- 2t-3 + 11t-2 - 27t-1 + 37 - 27t + 11t2 - 2t3

Conway Polynomial:

1 - z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a8, K11a38, K11a187, ...}

Determinant and Signature:

{117, 0}

Jones Polynomial:

- q-7 + 4q-6 - 7q-5 + 11q-4 - 16q-3 + 18q-2 - 18q-1 + 17 - 12q + 8q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 + q-20 + 2q-18 - 2q-16 + 2q-14 - 3q-10 + 2q-8 - 3q-6 + q-4 + q-2 + 5q2 - 3q4 + q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 + 2 - z2 - 2z4 - z6 - 2a2 - 3a2z2 - 2a2z4 - a2z6 + a4 + 3a4z2 + 2a4z4 - a6z2

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 + 2a-2z2 - 8a-2z4 + 8a-2z6 + a-1z3 - 11a-1z5 + 10a-1z7 + 2 - z2 + 3z4 - 14z6 + 10z8 - 6az3 + 10az5 - 15az7 + 8az9 + 2a2 - 12a2z2 + 35a2z4 - 33a2z6 + 5a2z8 + 3a2z10 - a3z - 18a3z3 + 53a3z5 - 49a3z7 + 14a3z9 + a4 - 12a4z2 + 37a4z4 - 26a4z6 - a4z8 + 3a4z10 - a5z - 8a5z3 + 25a5z5 - 23a5z7 + 6a5z9 - 3a6z2 + 14a6z4 - 15a6z6 + 4a6z8 + a7z3 - 3a7z5 + a7z7

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

{-1, 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 11249. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 7 3     j = 1               10 5       j = -1             9 8         j = -3           9 9           j = -5         7 9             j = -7       4 9               j = -9     3 7                 j = -11   1 4                   j = -13   3                     j = -15 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, 249]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 249]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[18, 8, 19, 7], > X[16, 9, 17, 10], X[22, 11, 1, 12], X[20, 13, 21, 14], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 249]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 249]]
Out[5]=	DTCode[6, 8, 10, 18, 16, 22, 20, 4, 2, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 249]][t]
Out[6]=	2 11 27 2 3 37 - -- + -- - -- - 27 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 249]][z]
Out[7]=	2 4 6 1 - z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 8], Knot[11, Alternating, 38], > Knot[11, Alternating, 187], Knot[11, Alternating, 249]}
In[9]:=	{KnotDet[Knot[11, Alternating, 249]], KnotSignature[Knot[11, Alternating, 249]]}
Out[9]=	{117, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 249]][q]
Out[10]=	-7 4 7 11 16 18 18 2 3 4 17 - q + -- - -- + -- - -- + -- - -- - 12 q + 8 q - 4 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 249]}
In[12]:=	A2Invariant[Knot[11, Alternating, 249]][q]
Out[12]=	-22 -20 2 2 2 3 2 3 -4 -2 2 4 6 -q + q + --- - --- + --- - --- + -- - -- + q + q + 5 q - 3 q + q + 18 16 14 10 8 6 q q q q q q 8 10 12 > q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 249]][a, z]
Out[13]=	2 4 2 4 2 z 2 2 4 2 6 2 4 z 2 4 2 - 2 a + a - z + -- - 3 a z + 3 a z - a z - 2 z + -- - 2 a z + 2 2 a a 4 4 6 2 6 > 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 249]][a, z]
Out[14]=	2 2 4 3 5 2 2 z 2 2 4 2 6 2 2 + 2 a + a - a z - a z - z + ---- - 12 a z - 12 a z - 3 a z - 2 a 3 3 4 4 2 z z 3 3 3 5 3 7 3 4 z 8 z > ---- + -- - 6 a z - 18 a z - 8 a z + a z + 3 z + -- - ---- + 3 a 4 2 a a a 5 5 2 4 4 4 6 4 4 z 11 z 5 3 5 > 35 a z + 37 a z + 14 a z + ---- - ----- + 10 a z + 53 a z + 3 a a 6 5 5 7 5 6 8 z 2 6 4 6 6 6 > 25 a z - 3 a z - 14 z + ---- - 33 a z - 26 a z - 15 a z + 2 a 7 10 z 7 3 7 5 7 7 7 8 2 8 4 8 > ----- - 15 a z - 49 a z - 23 a z + a z + 10 z + 5 a z - a z + a 6 8 9 3 9 5 9 2 10 4 10 > 4 a z + 8 a z + 14 a z + 6 a z + 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 249]], Vassiliev[3][Knot[11, Alternating, 249]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 249]][q, t]
Out[16]=	8 1 3 1 4 3 7 4 9 - + 10 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 7 9 9 9 9 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 5 q t + 7 q t + 3 q t + 5 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a249

K11a248 K11a250