10 28

10_27

10_29

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10 28 Quick Notes

10 28 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X11,1,12,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -8, 9, -3, 4, -7, 6, -5, 3, -9, 8
Dowker-Thistlethwaite code	4 10 14 16 2 20 18 8 6 12
Conway Notation	[31312]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-8] Hyperbolic Volume 10.2647 A-Polynomial See Data:10 28/A-polynomial

[edit Notes for 10 28's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant 0

[edit Notes for 10 28's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-13 t+19-13 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 53, 0 }
Jones polynomial	[math]\displaystyle{ -q^7+2 q^6-4 q^5+6 q^4-7 q^3+9 q^2-8 q+7-5 q^{-1} +3 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ 2 z^4 a^{-2} +z^4 a^{-4} +z^4-a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +3 a^{-2} - a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +5 z^8 a^{-4} +2 z^8 a^{-6} +5 z^7 a^{-1} +4 z^7 a^{-3} +z^7 a^{-7} -z^6 a^{-2} -16 z^6 a^{-4} -9 z^6 a^{-6} +6 z^6+5 a z^5-6 z^5 a^{-1} -18 z^5 a^{-3} -12 z^5 a^{-5} -5 z^5 a^{-7} +3 a^2 z^4-12 z^4 a^{-2} +11 z^4 a^{-4} +12 z^4 a^{-6} -8 z^4+a^3 z^3-4 a z^3-2 z^3 a^{-1} +13 z^3 a^{-3} +18 z^3 a^{-5} +8 z^3 a^{-7} -a^2 z^2+10 z^2 a^{-2} -5 z^2 a^{-6} +4 z^2+a z+z a^{-1} -2 z a^{-3} -6 z a^{-5} -4 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{10}+q^8+q^6-2 q^4+q^2-1+2 q^{-4} + q^{-6} +3 q^{-8} + q^{-14} -2 q^{-16} - q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+2 q^{44}-q^{42}-2 q^{40}+8 q^{38}-11 q^{36}+14 q^{34}-13 q^{32}+6 q^{30}-10 q^{26}+20 q^{24}-26 q^{22}+25 q^{20}-18 q^{18}+6 q^{16}+9 q^{14}-20 q^{12}+30 q^{10}-31 q^8+22 q^6-10 q^4-7 q^2+20-24 q^{-2} +21 q^{-4} -9 q^{-6} -5 q^{-8} +16 q^{-10} -21 q^{-12} +9 q^{-14} +10 q^{-16} -28 q^{-18} +38 q^{-20} -32 q^{-22} +12 q^{-24} +22 q^{-26} -45 q^{-28} +60 q^{-30} -53 q^{-32} +30 q^{-34} +7 q^{-36} -36 q^{-38} +56 q^{-40} -49 q^{-42} +34 q^{-44} -5 q^{-46} -18 q^{-48} +32 q^{-50} -29 q^{-52} +16 q^{-54} +4 q^{-56} -24 q^{-58} +29 q^{-60} -21 q^{-62} + q^{-64} +22 q^{-66} -40 q^{-68} +44 q^{-70} -34 q^{-72} +6 q^{-74} +20 q^{-76} -42 q^{-78} +47 q^{-80} -37 q^{-82} +15 q^{-84} +5 q^{-86} -22 q^{-88} +27 q^{-90} -23 q^{-92} +13 q^{-94} -2 q^{-96} -5 q^{-98} +6 q^{-100} -6 q^{-102} +4 q^{-104} - q^{-106} + q^{-108} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_28.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^7+2 q^5-2 q^3+2 q- q^{-1} + q^{-3} +2 q^{-5} - q^{-7} +2 q^{-9} -2 q^{-11} + q^{-13} - q^{-15} }[/math]
2	[math]\displaystyle{ q^{20}-2 q^{18}+4 q^{14}-6 q^{12}+q^{10}+6 q^8-9 q^6+3 q^4+8 q^2-8+8 q^{-4} -2 q^{-6} -5 q^{-8} +4 q^{-10} +5 q^{-12} -5 q^{-14} -3 q^{-16} +9 q^{-18} -3 q^{-20} -8 q^{-22} +9 q^{-24} + q^{-26} -10 q^{-28} +5 q^{-30} +4 q^{-32} -7 q^{-34} + q^{-36} +4 q^{-38} -2 q^{-40} - q^{-42} + q^{-44} }[/math]
3	[math]\displaystyle{ -q^{39}+2 q^{37}-2 q^{33}+3 q^{29}+q^{27}-6 q^{25}+2 q^{23}+4 q^{21}-5 q^{19}-5 q^{17}+10 q^{15}+5 q^{13}-16 q^{11}-5 q^9+21 q^7+8 q^5-23 q^3-14 q+23 q^{-1} +19 q^{-3} -12 q^{-5} -23 q^{-7} +2 q^{-9} +25 q^{-11} +14 q^{-13} -21 q^{-15} -21 q^{-17} +14 q^{-19} +29 q^{-21} -7 q^{-23} -32 q^{-25} - q^{-27} +28 q^{-29} +8 q^{-31} -28 q^{-33} -13 q^{-35} +22 q^{-37} +23 q^{-39} -19 q^{-41} -26 q^{-43} +10 q^{-45} +30 q^{-47} -2 q^{-49} -30 q^{-51} -9 q^{-53} +27 q^{-55} +16 q^{-57} -18 q^{-59} -21 q^{-61} +8 q^{-63} +21 q^{-65} -15 q^{-69} -6 q^{-71} +9 q^{-73} +7 q^{-75} -3 q^{-77} -5 q^{-79} +2 q^{-83} + q^{-85} - q^{-87} }[/math]
4	[math]\displaystyle{ q^{64}-2 q^{62}+2 q^{58}-2 q^{56}+3 q^{54}-5 q^{52}+2 q^{50}+4 q^{48}-6 q^{46}+11 q^{44}-8 q^{42}-q^{40}-q^{38}-13 q^{36}+28 q^{34}+2 q^{32}-7 q^{30}-25 q^{28}-28 q^{26}+59 q^{24}+36 q^{22}-13 q^{20}-72 q^{18}-65 q^{16}+86 q^{14}+97 q^{12}+12 q^{10}-110 q^8-126 q^6+64 q^4+136 q^2+81-73 q^{-2} -161 q^{-4} -26 q^{-6} +86 q^{-8} +129 q^{-10} +32 q^{-12} -103 q^{-14} -100 q^{-16} -34 q^{-18} +93 q^{-20} +121 q^{-22} +11 q^{-24} -101 q^{-26} -116 q^{-28} +17 q^{-30} +129 q^{-32} +85 q^{-34} -66 q^{-36} -130 q^{-38} -31 q^{-40} +103 q^{-42} +109 q^{-44} -45 q^{-46} -118 q^{-48} -54 q^{-50} +79 q^{-52} +128 q^{-54} -16 q^{-56} -105 q^{-58} -90 q^{-60} +37 q^{-62} +141 q^{-64} +41 q^{-66} -59 q^{-68} -124 q^{-70} -43 q^{-72} +108 q^{-74} +94 q^{-76} +28 q^{-78} -102 q^{-80} -111 q^{-82} +17 q^{-84} +78 q^{-86} +103 q^{-88} -13 q^{-90} -99 q^{-92} -60 q^{-94} - q^{-96} +92 q^{-98} +57 q^{-100} -21 q^{-102} -55 q^{-104} -58 q^{-106} +23 q^{-108} +48 q^{-110} +29 q^{-112} -4 q^{-114} -43 q^{-116} -16 q^{-118} +6 q^{-120} +20 q^{-122} +17 q^{-124} -9 q^{-126} -9 q^{-128} -7 q^{-130} + q^{-132} +7 q^{-134} + q^{-136} -2 q^{-140} - q^{-142} + q^{-144} }[/math]
5	[math]\displaystyle{ -q^{95}+2 q^{93}-2 q^{89}+2 q^{87}-q^{85}-q^{83}+2 q^{81}-3 q^{77}-q^{73}+6 q^{69}+8 q^{67}-18 q^{63}-17 q^{61}+3 q^{59}+27 q^{57}+42 q^{55}+2 q^{53}-64 q^{51}-72 q^{49}+5 q^{47}+100 q^{45}+122 q^{43}+9 q^{41}-167 q^{39}-205 q^{37}-17 q^{35}+244 q^{33}+303 q^{31}+56 q^{29}-318 q^{27}-446 q^{25}-132 q^{23}+379 q^{21}+595 q^{19}+251 q^{17}-382 q^{15}-720 q^{13}-422 q^{11}+304 q^9+801 q^7+602 q^5-146 q^3-774 q-739 q^{-1} -92 q^{-3} +627 q^{-5} +812 q^{-7} +332 q^{-9} -376 q^{-11} -743 q^{-13} -531 q^{-15} +64 q^{-17} +577 q^{-19} +630 q^{-21} +232 q^{-23} -322 q^{-25} -624 q^{-27} -447 q^{-29} +61 q^{-31} +525 q^{-33} +571 q^{-35} +150 q^{-37} -399 q^{-39} -599 q^{-41} -276 q^{-43} +276 q^{-45} +571 q^{-47} +335 q^{-49} -212 q^{-51} -536 q^{-53} -336 q^{-55} +185 q^{-57} +517 q^{-59} +337 q^{-61} -181 q^{-63} -538 q^{-65} -365 q^{-67} +185 q^{-69} +564 q^{-71} +418 q^{-73} -130 q^{-75} -590 q^{-77} -519 q^{-79} +46 q^{-81} +573 q^{-83} +600 q^{-85} +114 q^{-87} -487 q^{-89} -677 q^{-91} -285 q^{-93} +337 q^{-95} +670 q^{-97} +458 q^{-99} -116 q^{-101} -588 q^{-103} -571 q^{-105} -121 q^{-107} +405 q^{-109} +604 q^{-111} +328 q^{-113} -170 q^{-115} -515 q^{-117} -466 q^{-119} -81 q^{-121} +343 q^{-123} +487 q^{-125} +269 q^{-127} -113 q^{-129} -390 q^{-131} -375 q^{-133} -96 q^{-135} +226 q^{-137} +358 q^{-139} +231 q^{-141} -32 q^{-143} -252 q^{-145} -277 q^{-147} -111 q^{-149} +110 q^{-151} +224 q^{-153} +177 q^{-155} +25 q^{-157} -127 q^{-159} -171 q^{-161} -95 q^{-163} +28 q^{-165} +110 q^{-167} +107 q^{-169} +38 q^{-171} -43 q^{-173} -79 q^{-175} -58 q^{-177} -3 q^{-179} +38 q^{-181} +44 q^{-183} +25 q^{-185} -7 q^{-187} -26 q^{-189} -21 q^{-191} -4 q^{-193} +6 q^{-195} +11 q^{-197} +9 q^{-199} - q^{-201} -5 q^{-203} -3 q^{-205} - q^{-207} +2 q^{-211} + q^{-213} - q^{-215} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{10}+q^8+q^6-2 q^4+q^2-1+2 q^{-4} + q^{-6} +3 q^{-8} + q^{-14} -2 q^{-16} - q^{-22} }[/math]
1,1	[math]\displaystyle{ q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+20 q^{20}-32 q^{18}+42 q^{16}-50 q^{14}+62 q^{12}-78 q^{10}+82 q^8-86 q^6+92 q^4-94 q^2+86-70 q^{-2} +55 q^{-4} -24 q^{-6} -14 q^{-8} +66 q^{-10} -111 q^{-12} +172 q^{-14} -212 q^{-16} +246 q^{-18} -262 q^{-20} +258 q^{-22} -234 q^{-24} +184 q^{-26} -131 q^{-28} +70 q^{-30} -4 q^{-32} -60 q^{-34} +104 q^{-36} -136 q^{-38} +154 q^{-40} -156 q^{-42} +138 q^{-44} -112 q^{-46} +86 q^{-48} -58 q^{-50} +33 q^{-52} -18 q^{-54} +8 q^{-56} -2 q^{-58} + q^{-60} }[/math]
2,0	[math]\displaystyle{ q^{26}-q^{24}-2 q^{22}+2 q^{20}+3 q^{18}-2 q^{16}-5 q^{14}+3 q^{12}+4 q^{10}-7 q^8-4 q^6+8 q^4+4 q^2-5- q^{-2} +5 q^{-4} + q^{-6} -4 q^{-8} + q^{-10} +2 q^{-12} - q^{-14} +6 q^{-16} +5 q^{-18} - q^{-20} - q^{-22} +5 q^{-24} + q^{-26} -6 q^{-28} -4 q^{-30} +4 q^{-32} + q^{-34} -7 q^{-36} -3 q^{-38} +3 q^{-40} +2 q^{-42} -3 q^{-44} -2 q^{-46} +3 q^{-48} +2 q^{-50} - q^{-52} - q^{-54} + q^{-58} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{22}-2 q^{20}-q^{18}+5 q^{16}-3 q^{14}-4 q^{12}+8 q^{10}-2 q^8-8 q^6+7 q^4-8+5 q^{-2} +3 q^{-4} -4 q^{-6} +2 q^{-8} +5 q^{-10} +4 q^{-12} -3 q^{-14} +4 q^{-16} +8 q^{-18} -5 q^{-20} +7 q^{-24} -7 q^{-26} -3 q^{-28} +3 q^{-30} -7 q^{-32} -2 q^{-34} +3 q^{-36} -3 q^{-38} + q^{-40} +2 q^{-42} - q^{-44} + q^{-46} }[/math]
1,0,0	[math]\displaystyle{ -q^{13}+q^{11}+q^7-2 q^5+q^3-2 q+2 q^{-5} +2 q^{-7} +2 q^{-9} +3 q^{-11} + q^{-15} - q^{-17} + q^{-19} -2 q^{-21} - q^{-25} - q^{-29} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{22}+2 q^{20}-3 q^{18}+5 q^{16}-7 q^{14}+8 q^{12}-10 q^{10}+10 q^8-10 q^6+9 q^4-6 q^2+2+3 q^{-2} -7 q^{-4} +14 q^{-6} -16 q^{-8} +21 q^{-10} -20 q^{-12} +21 q^{-14} -18 q^{-16} +14 q^{-18} -9 q^{-20} +4 q^{-22} + q^{-24} -5 q^{-26} +9 q^{-28} -11 q^{-30} +11 q^{-32} -10 q^{-34} +9 q^{-36} -7 q^{-38} +5 q^{-40} -4 q^{-42} + q^{-44} - q^{-46} }[/math]
1,0	[math]\displaystyle{ q^{36}-2 q^{32}-2 q^{30}+q^{28}+5 q^{26}+2 q^{24}-5 q^{22}-6 q^{20}+q^{18}+9 q^{16}+4 q^{14}-7 q^{12}-9 q^{10}+q^8+10 q^6+3 q^4-8 q^2-7+5 q^{-2} +9 q^{-4} - q^{-6} -9 q^{-8} - q^{-10} +8 q^{-12} +5 q^{-14} -5 q^{-16} -3 q^{-18} +5 q^{-20} +6 q^{-22} -3 q^{-24} -4 q^{-26} +5 q^{-28} +9 q^{-30} - q^{-32} -10 q^{-34} -3 q^{-36} +10 q^{-38} +8 q^{-40} -6 q^{-42} -13 q^{-44} - q^{-46} +10 q^{-48} +4 q^{-50} -8 q^{-52} -9 q^{-54} +2 q^{-56} +7 q^{-58} + q^{-60} -5 q^{-62} -3 q^{-64} +3 q^{-66} +3 q^{-68} - q^{-70} - q^{-72} + q^{-76} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+2 q^{44}-q^{42}-2 q^{40}+8 q^{38}-11 q^{36}+14 q^{34}-13 q^{32}+6 q^{30}-10 q^{26}+20 q^{24}-26 q^{22}+25 q^{20}-18 q^{18}+6 q^{16}+9 q^{14}-20 q^{12}+30 q^{10}-31 q^8+22 q^6-10 q^4-7 q^2+20-24 q^{-2} +21 q^{-4} -9 q^{-6} -5 q^{-8} +16 q^{-10} -21 q^{-12} +9 q^{-14} +10 q^{-16} -28 q^{-18} +38 q^{-20} -32 q^{-22} +12 q^{-24} +22 q^{-26} -45 q^{-28} +60 q^{-30} -53 q^{-32} +30 q^{-34} +7 q^{-36} -36 q^{-38} +56 q^{-40} -49 q^{-42} +34 q^{-44} -5 q^{-46} -18 q^{-48} +32 q^{-50} -29 q^{-52} +16 q^{-54} +4 q^{-56} -24 q^{-58} +29 q^{-60} -21 q^{-62} + q^{-64} +22 q^{-66} -40 q^{-68} +44 q^{-70} -34 q^{-72} +6 q^{-74} +20 q^{-76} -42 q^{-78} +47 q^{-80} -37 q^{-82} +15 q^{-84} +5 q^{-86} -22 q^{-88} +27 q^{-90} -23 q^{-92} +13 q^{-94} -2 q^{-96} -5 q^{-98} +6 q^{-100} -6 q^{-102} +4 q^{-104} - q^{-106} + q^{-108} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 28"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ 4 t^2-13 t+19-13 t^{-1} +4 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 z^4+3 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 53, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^7+2 q^6-4 q^5+6 q^4-7 q^3+9 q^2-8 q+7-5 q^{-1} +3 q^{-2} - q^{-3} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ 2 z^4 a^{-2} +z^4 a^{-4} +z^4-a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +3 a^{-2} - a^{-6} -1 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +5 z^8 a^{-4} +2 z^8 a^{-6} +5 z^7 a^{-1} +4 z^7 a^{-3} +z^7 a^{-7} -z^6 a^{-2} -16 z^6 a^{-4} -9 z^6 a^{-6} +6 z^6+5 a z^5-6 z^5 a^{-1} -18 z^5 a^{-3} -12 z^5 a^{-5} -5 z^5 a^{-7} +3 a^2 z^4-12 z^4 a^{-2} +11 z^4 a^{-4} +12 z^4 a^{-6} -8 z^4+a^3 z^3-4 a z^3-2 z^3 a^{-1} +13 z^3 a^{-3} +18 z^3 a^{-5} +8 z^3 a^{-7} -a^2 z^2+10 z^2 a^{-2} -5 z^2 a^{-6} +4 z^2+a z+z a^{-1} -2 z a^{-3} -6 z a^{-5} -4 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]

Vassiliev invariants

V2 and V3:

(3, 4)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 12 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 126 }[/math] [math]\displaystyle{ 2 }[/math] [math]\displaystyle{ 384 }[/math] [math]\displaystyle{ \frac{1760}{3} }[/math] [math]\displaystyle{ \frac{320}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ 1512 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{27311}{10} }[/math] [math]\displaystyle{ \frac{6694}{15} }[/math] [math]\displaystyle{ \frac{2314}{5} }[/math] [math]\displaystyle{ \frac{91}{2} }[/math] [math]\displaystyle{ -\frac{529}{10} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-3	-2	-1	0	1	2	3	4	5	6	7	χ
15											1	-1
13										1		1
11									3	1		-2
9								3	1			2
7							4	3				-1
5						5	3					2
3					3	4						1
1				4	5							-1
-1			2	4								2
-3		1	3									-2
-5		2										2
-7	1											-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 28]]
Out[2]=	10
In[3]:=	PD[Knot[10, 28]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[13, 19, 14, 18], X[5, 15, 6, 14], X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8], X[11, 1, 12, 20], X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 28]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -8, 9, -3, 4, -7, 6, -5, 3, -9, 8]
In[5]:=	BR[Knot[10, 28]]
Out[5]=	BR[5, {1, 1, 2, -1, 2, 2, 3, -2, -4, 3, -4, -4}]
In[6]:=	alex = Alexander[Knot[10, 28]][t]
Out[6]=	4 13 2 19 + -- - -- - 13 t + 4 t 2 t t
In[7]:=	Conway[Knot[10, 28]][z]
Out[7]=	2 4 1 + 3 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 28], Knot[10, 37]}
In[9]:=	{KnotDet[Knot[10, 28]], KnotSignature[Knot[10, 28]]}
Out[9]=	{53, 0}
In[10]:=	J=Jones[Knot[10, 28]][q]
Out[10]=	-3 3 5 2 3 4 5 6 7 7 - q + -- - - - 8 q + 9 q - 7 q + 6 q - 4 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 28]}
In[12]:=	A2Invariant[Knot[10, 28]][q]
Out[12]=	-10 -8 -6 2 -2 4 6 8 14 16 22 -1 - q + q + q - -- + q + 2 q + q + 3 q + q - 2 q - q 4 q
In[13]:=	Kauffman[Knot[10, 28]][a, z]
Out[13]=	2 2 -6 3 4 z 6 z 2 z z 2 5 z 10 z -1 + a - -- - --- - --- - --- + - + a z + 4 z - ---- + ----- - 2 7 5 3 a 6 2 a a a a a a 3 3 3 3 4 2 2 8 z 18 z 13 z 2 z 3 3 3 4 12 z a z + ---- + ----- + ----- - ---- - 4 a z + a z - 8 z + ----- + 7 5 3 a 6 a a a a 4 4 5 5 5 5 11 z 12 z 2 4 5 z 12 z 18 z 6 z 5 ----- - ----- + 3 a z - ---- - ----- - ----- - ---- + 5 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 8 8 8 6 9 z 16 z z z 4 z 5 z 2 z 5 z 3 z 6 z - ---- - ----- - -- + -- + ---- + ---- + ---- + ---- + ---- + 6 4 2 7 3 a 6 4 2 a a a a a a a a 9 9 z z -- + -- 5 3 a a
In[14]:=	{Vassiliev[2][Knot[10, 28]], Vassiliev[3][Knot[10, 28]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[10, 28]][q, t]
Out[15]=	4 1 2 1 3 2 3 - + 4 q + ----- + ----- + ----- + ---- + --- + 5 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 4 q t + 5 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 11 5 11 6 13 6 15 7 3 q t + q t + q t + q t