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

K11a281 K11a283

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 14 4 18 20 22 2 12 8

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 26t-1 - 29 + 26t - 16t2 + 6t3 - t4

Conway Polynomial:

1 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a81, ...}

Determinant and Signature:

{127, -2}

Jones Polynomial:

q-7 - 4q-6 + 8q-5 - 13q-4 + 18q-3 - 20q-2 + 20q-1 - 17 + 13q - 8q2 + 4q3 - q4

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

{K11a81, ...}

A2 (sl(3)) Invariant:

q-20 - 2q-18 + 2q-16 - 2q-14 - q-12 + 3q-10 - 3q-8 + 5q-6 - 2q-4 + q-2 + 1 - 3q2 + 3q4 - q6 + q8 + q10 - q12

HOMFLY-PT Polynomial:

- 2a-2z2 - a-2z4 + 1 + 6z2 + 7z4 + 2z6 - 6a2z2 - 9a2z4 - 5a2z6 - a2z8 + 2a4z2 + 3a4z4 + a4z6

Kauffman Polynomial:

2a-3z3 - 3a-3z5 + a-3z7 - 5a-2z2 + 14a-2z4 - 14a-2z6 + 4a-2z8 - 5a-1z3 + 18a-1z5 - 20a-1z7 + 6a-1z9 + 1 - 14z2 + 41z4 - 33z6 + 2z8 + 3z10 - 9az3 + 39az5 - 46az7 + 15az9 - 12a2z2 + 45a2z4 - 47a2z6 + 10a2z8 + 3a2z10 + 3a3z3 + a3z5 - 14a3z7 + 9a3z9 - 2a4z2 + 10a4z4 - 20a4z6 + 12a4z8 + 3a5z3 - 13a5z5 + 11a5z7 + a6z2 - 7a6z4 + 8a6z6 - 2a7z3 + 4a7z5 + a8z4

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

{0, 0}

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 11282. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 8 3     j = 1               9 5       j = -1             11 8         j = -3           10 10           j = -5         8 10             j = -7       5 10               j = -9     3 8                 j = -11   1 5                   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, 282]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 282]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[4, 10, 5, 9], X[18, 11, 19, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[2, 17, 3, 18], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 282]]
Out[4]=	GaussCode[1, -9, 2, -5, 3, -1, 4, -11, 5, -2, 6, -10, 7, -4, 8, -3, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 282]]
Out[5]=	DTCode[6, 10, 16, 14, 4, 18, 20, 22, 2, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 282]][t]
Out[6]=	-4 6 16 26 2 3 4 -29 - t + -- - -- + -- + 26 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 282]][z]
Out[7]=	6 8 1 - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 81], Knot[11, Alternating, 282]}
In[9]:=	{KnotDet[Knot[11, Alternating, 282]], KnotSignature[Knot[11, Alternating, 282]]}
Out[9]=	{127, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 282]][q]
Out[10]=	-7 4 8 13 18 20 20 2 3 4 -17 + q - -- + -- - -- + -- - -- + -- + 13 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, 81], Knot[11, Alternating, 282]}
In[12]:=	A2Invariant[Knot[11, Alternating, 282]][q]
Out[12]=	-20 2 2 2 -12 3 3 5 2 -2 2 4 1 + q - --- + --- - --- - q + --- - -- + -- - -- + q - 3 q + 3 q - 18 16 14 10 8 6 4 q q q q q q q 6 8 10 12 > q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 282]][a, z]
Out[13]=	2 4 2 2 z 2 2 4 2 4 z 2 4 4 4 6 1 + 6 z - ---- - 6 a z + 2 a z + 7 z - -- - 9 a z + 3 a z + 2 z - 2 2 a a 2 6 4 6 2 8 > 5 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 282]][a, z]
Out[14]=	2 3 3 2 5 z 2 2 4 2 6 2 2 z 5 z 3 1 - 14 z - ---- - 12 a z - 2 a z + a z + ---- - ---- - 9 a z + 2 3 a a a 4 3 3 5 3 7 3 4 14 z 2 4 4 4 > 3 a z + 3 a z - 2 a z + 41 z + ----- + 45 a z + 10 a z - 2 a 5 5 6 4 8 4 3 z 18 z 5 3 5 5 5 7 5 > 7 a z + a z - ---- + ----- + 39 a z + a z - 13 a z + 4 a z - 3 a a 6 7 7 6 14 z 2 6 4 6 6 6 z 20 z 7 > 33 z - ----- - 47 a z - 20 a z + 8 a z + -- - ----- - 46 a z - 2 3 a a a 8 9 3 7 5 7 8 4 z 2 8 4 8 6 z 9 > 14 a z + 11 a z + 2 z + ---- + 10 a z + 12 a z + ---- + 15 a z + 2 a a 3 9 10 2 10 > 9 a z + 3 z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 282]], Vassiliev[3][Knot[11, Alternating, 282]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 282]][q, t]
Out[16]=	10 11 1 3 1 5 3 8 5 10 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 8 10 10 8 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 9 q t + 5 q t + 8 q t + 3 q t + 5 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a282

K11a281 K11a283