9 48

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9 47.gif

9_47

9 49.gif

9_49

9 48.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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[edit 9 48 Quick Notes]
[edit 9 48 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code -1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7
Dowker-Thistlethwaite code 4 10 -14 -12 16 2 -6 18 8
Conway Notation [21,21,21-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

9 48 ML.gif 9 48 AP.gif
[{10, 4}, {5, 3}, {4, 7}, {2, 5}, {8, 6}, {7, 1}, {3, 8}, {9, 2}, {6, 10}, {1, 9}]

[edit Notes on presentations of 9 48]


Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["9 48"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 X15,18,16,1 X9,17,10,16 X17,9,18,8
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7
In[6]:= DTCode[K]
Out[6]= 4 10 -14 -12 16 2 -6 18 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [21,21,21-]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(4,\{1,1,2,-1,2,1,-3,2,-1,2,-3\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 4, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
9 48 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 4}, {5, 3}, {4, 7}, {2, 5}, {8, 6}, {7, 1}, {3, 8}, {9, 2}, {6, 10}, {1, 9}]
In[14]:= Draw[ap]
9 48 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 2
Maximal Thurston-Bennequin number [-1][-8]
Hyperbolic Volume 9.53188
A-Polynomial See Data:9 48/A-polynomial

[edit Notes for 9 48'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 2

[edit Notes for 9 48's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^2+7 t-11+7 t^{-1} - t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature { 27, 2 }
Jones polynomial [math]\displaystyle{ -2 q^6+3 q^5-4 q^4+6 q^3-4 q^2+4 q-3+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4 a^{-2} -z^2 a^{-2} +3 z^2 a^{-4} +z^2+3 a^{-4} -2 a^{-6} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +3 z^6 a^{-2} +4 z^6 a^{-4} +z^6 a^{-6} +3 z^5 a^{-1} +2 z^5 a^{-3} -z^5 a^{-5} -5 z^4 a^{-2} -6 z^4 a^{-4} +z^4-5 z^3 a^{-1} -3 z^3 a^{-3} +5 z^3 a^{-5} +3 z^3 a^{-7} +2 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -z^2-z a^{-3} -5 z a^{-5} -4 z a^{-7} +3 a^{-4} +2 a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ q^4-q^2-1+ q^{-2} - q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} + q^{-16} -2 q^{-18} -2 q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+3 q^{10}+q^8-6 q^6+14 q^4-14 q^2+15-8 q^{-2} -7 q^{-4} +16 q^{-6} -23 q^{-8} +18 q^{-10} -9 q^{-12} -5 q^{-14} +13 q^{-16} -13 q^{-18} +10 q^{-20} + q^{-22} -13 q^{-24} +16 q^{-26} -13 q^{-28} + q^{-30} +15 q^{-32} -24 q^{-34} +27 q^{-36} -14 q^{-38} +11 q^{-40} +7 q^{-42} -22 q^{-44} +29 q^{-46} -22 q^{-48} +19 q^{-50} -2 q^{-52} -14 q^{-54} +21 q^{-56} -8 q^{-58} +6 q^{-60} + q^{-62} -15 q^{-64} +14 q^{-66} -5 q^{-68} -8 q^{-70} +14 q^{-72} -24 q^{-74} +21 q^{-76} -5 q^{-78} -10 q^{-80} +11 q^{-82} -18 q^{-84} +16 q^{-86} -9 q^{-88} -2 q^{-90} +3 q^{-92} -7 q^{-94} +7 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_48.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^3-2 q+ q^{-1} +2 q^{-5} +2 q^{-7} - q^{-9} + q^{-11} -2 q^{-13} }[/math]
2 [math]\displaystyle{ q^{10}-2 q^8-2 q^6+6 q^4-6+5 q^{-2} +2 q^{-4} -7 q^{-6} +2 q^{-8} +4 q^{-10} -2 q^{-12} + q^{-14} +4 q^{-16} +4 q^{-18} -5 q^{-20} +6 q^{-24} -7 q^{-26} -4 q^{-28} +5 q^{-30} -3 q^{-32} -3 q^{-34} +3 q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ q^{21}-2 q^{19}-2 q^{17}+3 q^{15}+6 q^{13}-13 q^9-4 q^7+14 q^5+11 q^3-13 q-19 q^{-1} +12 q^{-3} +26 q^{-5} -4 q^{-7} -27 q^{-9} - q^{-11} +23 q^{-13} +7 q^{-15} -22 q^{-17} -7 q^{-19} +14 q^{-21} +11 q^{-23} -7 q^{-25} -9 q^{-27} +2 q^{-29} +14 q^{-31} +9 q^{-33} -15 q^{-35} -12 q^{-37} +12 q^{-39} +21 q^{-41} -15 q^{-43} -27 q^{-45} +4 q^{-47} +25 q^{-49} -2 q^{-51} -24 q^{-53} -6 q^{-55} +19 q^{-57} +9 q^{-59} -9 q^{-61} -7 q^{-63} +5 q^{-65} +8 q^{-67} - q^{-69} -2 q^{-71} -2 q^{-73} }[/math]
4 [math]\displaystyle{ q^{36}-2 q^{34}-2 q^{32}+3 q^{30}+3 q^{28}+6 q^{26}-7 q^{24}-13 q^{22}-4 q^{20}+5 q^{18}+32 q^{16}+8 q^{14}-25 q^{12}-34 q^{10}-19 q^8+52 q^6+51 q^4-63-81 q^{-2} +30 q^{-4} +93 q^{-6} +64 q^{-8} -44 q^{-10} -120 q^{-12} -27 q^{-14} +78 q^{-16} +104 q^{-18} +2 q^{-20} -106 q^{-22} -62 q^{-24} +33 q^{-26} +90 q^{-28} +31 q^{-30} -60 q^{-32} -57 q^{-34} - q^{-36} +53 q^{-38} +38 q^{-40} -14 q^{-42} -43 q^{-44} -28 q^{-46} +20 q^{-48} +50 q^{-50} +35 q^{-52} -42 q^{-54} -62 q^{-56} -14 q^{-58} +60 q^{-60} +82 q^{-62} -29 q^{-64} -94 q^{-66} -63 q^{-68} +48 q^{-70} +120 q^{-72} +14 q^{-74} -85 q^{-76} -97 q^{-78} +5 q^{-80} +110 q^{-82} +57 q^{-84} -27 q^{-86} -84 q^{-88} -39 q^{-90} +52 q^{-92} +55 q^{-94} +21 q^{-96} -34 q^{-98} -40 q^{-100} + q^{-102} +17 q^{-104} +20 q^{-106} - q^{-108} -13 q^{-110} -7 q^{-112} -3 q^{-114} +3 q^{-116} +3 q^{-118} + q^{-120} }[/math]
5 [math]\displaystyle{ q^{55}-2 q^{53}-2 q^{51}+3 q^{49}+3 q^{47}+3 q^{45}-q^{43}-7 q^{41}-13 q^{39}-4 q^{37}+14 q^{35}+23 q^{33}+20 q^{31}-4 q^{29}-37 q^{27}-57 q^{25}-18 q^{23}+48 q^{21}+88 q^{19}+69 q^{17}-24 q^{15}-128 q^{13}-149 q^{11}-29 q^9+144 q^7+229 q^5+139 q^3-105 q-305 q^{-1} -269 q^{-3} +18 q^{-5} +328 q^{-7} +385 q^{-9} +115 q^{-11} -291 q^{-13} -471 q^{-15} -249 q^{-17} +206 q^{-19} +491 q^{-21} +361 q^{-23} -85 q^{-25} -460 q^{-27} -411 q^{-29} -18 q^{-31} +372 q^{-33} +424 q^{-35} +98 q^{-37} -283 q^{-39} -372 q^{-41} -141 q^{-43} +180 q^{-45} +312 q^{-47} +159 q^{-49} -111 q^{-51} -235 q^{-53} -155 q^{-55} +37 q^{-57} +179 q^{-59} +157 q^{-61} +10 q^{-63} -131 q^{-65} -166 q^{-67} -64 q^{-69} +96 q^{-71} +196 q^{-73} +133 q^{-75} -74 q^{-77} -229 q^{-79} -196 q^{-81} +25 q^{-83} +265 q^{-85} +295 q^{-87} +19 q^{-89} -300 q^{-91} -372 q^{-93} -115 q^{-95} +286 q^{-97} +454 q^{-99} +203 q^{-101} -245 q^{-103} -491 q^{-105} -306 q^{-107} +150 q^{-109} +483 q^{-111} +382 q^{-113} -30 q^{-115} -399 q^{-117} -420 q^{-119} -84 q^{-121} +288 q^{-123} +395 q^{-125} +182 q^{-127} -139 q^{-129} -317 q^{-131} -223 q^{-133} +23 q^{-135} +206 q^{-137} +204 q^{-139} +57 q^{-141} -98 q^{-143} -153 q^{-145} -90 q^{-147} +23 q^{-149} +79 q^{-151} +67 q^{-153} +19 q^{-155} -27 q^{-157} -45 q^{-159} -21 q^{-161} +3 q^{-163} +12 q^{-165} +17 q^{-167} +8 q^{-169} - q^{-171} -4 q^{-173} -2 q^{-175} -2 q^{-177} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^4-q^2-1+ q^{-2} - q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} + q^{-16} -2 q^{-18} -2 q^{-20} }[/math]
1,1 [math]\displaystyle{ q^{12}-4 q^{10}+10 q^8-20 q^6+30 q^4-42 q^2+58-62 q^{-2} +59 q^{-4} -48 q^{-6} +26 q^{-8} -2 q^{-10} -34 q^{-12} +68 q^{-14} -86 q^{-16} +112 q^{-18} -108 q^{-20} +114 q^{-22} -94 q^{-24} +70 q^{-26} -40 q^{-28} +2 q^{-30} +22 q^{-32} -46 q^{-34} +61 q^{-36} -68 q^{-38} +56 q^{-40} -48 q^{-42} +31 q^{-44} -22 q^{-46} +10 q^{-48} -2 q^{-50} +2 q^{-52} +2 q^{-54} }[/math]
2,0 [math]\displaystyle{ q^{12}-q^{10}-2 q^8+3 q^4+3 q^2-4- q^{-2} +4 q^{-4} + q^{-6} -4 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} +2 q^{-18} +5 q^{-20} +2 q^{-22} +6 q^{-24} +4 q^{-26} - q^{-28} + q^{-30} +3 q^{-32} -2 q^{-34} -7 q^{-36} -4 q^{-38} - q^{-40} -4 q^{-42} -6 q^{-44} +4 q^{-48} +4 q^{-50} + q^{-52} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^8-2 q^6+3 q^2-4+3 q^{-2} +4 q^{-4} -6 q^{-6} + q^{-10} -4 q^{-12} +6 q^{-16} +6 q^{-18} +7 q^{-20} +4 q^{-22} +6 q^{-24} -3 q^{-26} -9 q^{-28} - q^{-30} -6 q^{-32} -7 q^{-34} +4 q^{-36} + q^{-38} - q^{-40} +3 q^{-42} }[/math]
1,0,0 [math]\displaystyle{ q^5-q^3- q^{-1} + q^{-3} - q^{-5} + q^{-7} + q^{-9} + q^{-11} +2 q^{-13} +2 q^{-15} +3 q^{-17} + q^{-21} -2 q^{-23} -2 q^{-25} -2 q^{-27} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{10}-2 q^8+2 q^6-3 q^4+4 q^2-4+4 q^{-2} -3 q^{-4} +4 q^{-6} -2 q^{-8} - q^{-10} -2 q^{-14} +2 q^{-16} -6 q^{-18} +6 q^{-20} -3 q^{-22} +12 q^{-24} - q^{-26} +12 q^{-28} +10 q^{-32} - q^{-34} -5 q^{-38} -6 q^{-40} -2 q^{-42} -8 q^{-44} - q^{-46} -6 q^{-48} +4 q^{-50} -2 q^{-52} +4 q^{-54} -2 q^{-56} +3 q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+3 q^{10}+q^8-6 q^6+14 q^4-14 q^2+15-8 q^{-2} -7 q^{-4} +16 q^{-6} -23 q^{-8} +18 q^{-10} -9 q^{-12} -5 q^{-14} +13 q^{-16} -13 q^{-18} +10 q^{-20} + q^{-22} -13 q^{-24} +16 q^{-26} -13 q^{-28} + q^{-30} +15 q^{-32} -24 q^{-34} +27 q^{-36} -14 q^{-38} +11 q^{-40} +7 q^{-42} -22 q^{-44} +29 q^{-46} -22 q^{-48} +19 q^{-50} -2 q^{-52} -14 q^{-54} +21 q^{-56} -8 q^{-58} +6 q^{-60} + q^{-62} -15 q^{-64} +14 q^{-66} -5 q^{-68} -8 q^{-70} +14 q^{-72} -24 q^{-74} +21 q^{-76} -5 q^{-78} -10 q^{-80} +11 q^{-82} -18 q^{-84} +16 q^{-86} -9 q^{-88} -2 q^{-90} +3 q^{-92} -7 q^{-94} +7 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }[/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["9 48"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^2+7 t-11+7 t^{-1} - t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -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{ \{3,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 27, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -2 q^6+3 q^5-4 q^4+6 q^3-4 q^2+4 q-3+ q^{-1} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4 a^{-2} -z^2 a^{-2} +3 z^2 a^{-4} +z^2+3 a^{-4} -2 a^{-6} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +3 z^6 a^{-2} +4 z^6 a^{-4} +z^6 a^{-6} +3 z^5 a^{-1} +2 z^5 a^{-3} -z^5 a^{-5} -5 z^4 a^{-2} -6 z^4 a^{-4} +z^4-5 z^3 a^{-1} -3 z^3 a^{-3} +5 z^3 a^{-5} +3 z^3 a^{-7} +2 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -z^2-z a^{-3} -5 z a^{-5} -4 z a^{-7} +3 a^{-4} +2 a^{-6} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n1,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 48"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ -t^2+7 t-11+7 t^{-1} - t^{-2} }[/math], [math]\displaystyle{ -2 q^6+3 q^5-4 q^4+6 q^3-4 q^2+4 q-3+ q^{-1} }[/math] }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {K11n1,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (3, 5)
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{ 40 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 190 }[/math] [math]\displaystyle{ 42 }[/math] [math]\displaystyle{ 480 }[/math] [math]\displaystyle{ \frac{2800}{3} }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ 2280 }[/math] [math]\displaystyle{ 504 }[/math] [math]\displaystyle{ \frac{46031}{10} }[/math] [math]\displaystyle{ -\frac{6226}{15} }[/math] [math]\displaystyle{ \frac{38422}{15} }[/math] [math]\displaystyle{ \frac{337}{6} }[/math] [math]\displaystyle{ \frac{4111}{10} }[/math]

V2,1 through V6,9 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]2 is the signature of 9 48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-1012345χ
13       2-2
11      1 1
9     32 -1
7    31  2
5   13   2
3  33    0
1 12     1
-1 2      -2
-31       1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{18}+2 q^{17}-6 q^{16}+q^{15}+10 q^{14}-15 q^{13}-2 q^{12}+23 q^{11}-21 q^{10}-7 q^9+32 q^8-21 q^7-10 q^6+29 q^5-15 q^4-12 q^3+20 q^2-6 q-9+9 q^{-1} -3 q^{-3} + q^{-4} }[/math]
3 [math]\displaystyle{ -2 q^{35}+q^{33}+9 q^{32}-5 q^{31}-12 q^{30}-q^{29}+27 q^{28}+5 q^{27}-37 q^{26}-19 q^{25}+49 q^{24}+32 q^{23}-58 q^{22}-50 q^{21}+61 q^{20}+68 q^{19}-67 q^{18}-74 q^{17}+58 q^{16}+92 q^{15}-62 q^{14}-86 q^{13}+47 q^{12}+94 q^{11}-44 q^{10}-83 q^9+26 q^8+79 q^7-15 q^6-67 q^5+2 q^4+53 q^3+8 q^2-37 q-12+22 q^{-1} +14 q^{-2} -13 q^{-3} -9 q^{-4} +4 q^{-5} +5 q^{-6} -3 q^{-8} + q^{-9} }[/math]
4 [math]\displaystyle{ q^{58}+2 q^{57}-6 q^{55}-4 q^{54}-5 q^{53}+14 q^{52}+21 q^{51}-9 q^{50}-20 q^{49}-46 q^{48}+20 q^{47}+76 q^{46}+25 q^{45}-23 q^{44}-137 q^{43}-25 q^{42}+133 q^{41}+109 q^{40}+30 q^{39}-242 q^{38}-127 q^{37}+145 q^{36}+208 q^{35}+136 q^{34}-314 q^{33}-238 q^{32}+114 q^{31}+273 q^{30}+247 q^{29}-336 q^{28}-312 q^{27}+66 q^{26}+293 q^{25}+324 q^{24}-321 q^{23}-342 q^{22}+18 q^{21}+278 q^{20}+353 q^{19}-269 q^{18}-327 q^{17}-36 q^{16}+222 q^{15}+350 q^{14}-178 q^{13}-268 q^{12}-93 q^{11}+127 q^{10}+306 q^9-70 q^8-166 q^7-119 q^6+22 q^5+213 q^4+6 q^3-58 q^2-90 q-41+102 q^{-1} +24 q^{-2} +5 q^{-3} -39 q^{-4} -40 q^{-5} +31 q^{-6} +9 q^{-7} +14 q^{-8} -6 q^{-9} -16 q^{-10} +4 q^{-11} +5 q^{-13} -3 q^{-15} + q^{-16} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

9 47.gif

9_47

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9_49

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